https://wiki.math.wisc.edu/api.php?action=feedcontributions&user=Cwu367&feedformat=atomUW-Math Wiki - User contributions [en]2022-09-24T19:39:58ZUser contributionsMediaWiki 1.35.6https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar&diff=23336Dynamics Seminar2022-07-17T15:01:55Z<p>Cwu367: </p>
<hr />
<div>The [[Dynamics]] seminar meets in room '''B309 of Van Vleck Hall''' on '''Mondays''' from '''2:30pm - 3:20pm'''. To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu. For more information, contact Paul Apisa, Marissa Loving, Caglar Uyanik, or Chenxi Wu. <br />
<br />
<br />
<br />
== Fall 2022 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
<br />
|-<br />
|September 12<br />
|[https://math.ou.edu/~jing/ Jing Tao] (OU)<br />
|[[# Jing Tao (OU) | ''TBA'']]<br />
|Dymarz and Uyanik<br />
|-<br />
|September 19<br />
|[https://math.temple.edu/~tug67058/ Rebekah Palmer] (Temple)[VIRTUAL]<br />
|[[# Rebekah Palmer (Temple)| ''TBA'']]<br />
|Loving<br />
|-<br />
|September 26<br />
|TBA<br />
|[[# TBA | ''TBA'']]<br />
|<br />
<br />
|-<br />
|October 3<br />
|[https://sites.google.com/view/beibei-liu/ Beibei Liu] (MIT)<br />
|[[# Beibei Liu (Georgia Tech) | ''TBA'']]<br />
| Dymarz<br />
|-<br />
|October 10<br />
|[https://mutanguha.com/ Jean Pierre Mutanguha] (Princeton)<br />
|[[# Jean Pierre Mutanguha (Princeton) | ''TBA'']]<br />
|Uyanik<br />
|-<br />
|October 17<br />
|[https://sites.google.com/ucsd.edu/ans032/ Anthony Sanchez] (UCSD)<br />
|[[# Anthony Sanchez (UCSD)| ''TBA'']]<br />
|Uyanik<br />
|-<br />
|October 24<br />
|[https://you.stonybrook.edu/aerchenko/ Alena Erchenko] (Stony Brook)<br />
|[[# Alena Erchenko| ''TBA'']]<br />
|Uyanik and Work<br />
|-<br />
|October 31<br />
|TBA<br />
|[[# TBA| ''TBA'']]<br />
|<br />
|-<br />
|November 7<br />
|[https://sites.google.com/bc.edu/ethan-farber/about-me?authuser=0/ Ethan Farber] (BC)<br />
|[[# Ethan Farber (BC)| ''TBA'']]<br />
|Loving<br />
|-<br />
|November 14<br />
|TBA<br />
|[[# TBA| ''TBA'']]<br />
|<br />
|-<br />
|November 21<br />
|Harry Baik (KAIST)<br />
|[[# TBA| ''TBA'']]<br />
|Wu<br />
|-<br />
|November 28<br />
|TBA<br />
|[[# TBA| ''TBA'']]<br />
|<br />
|-<br />
|December 5<br />
|TBA<br />
|[[# TBA| ''TBA'']]<br />
|<br />
|-<br />
|December 12<br />
|TBA<br />
|[[# TBA| ''TBA'']]<br />
|<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Jing Tao===<br />
<br />
===Rebekah Palmer===<br />
<br />
===Beibei Liu===<br />
<br />
===Jean Pierre Mutanguha===<br />
<br />
===Anthony Sanchez===<br />
<br />
===Alena Erchenko===<br />
<br />
===Ethan Farber===<br />
<br />
== Spring 2023 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 30<br />
|TBA<br />
|[[TBA| ''TBA'']]<br />
|<br />
|-<br />
|March 27<br />
|[https://www.carolynrabbott.com/ Carolyn Abbott] (Brandeis)<br />
|[[# Carolyn Abbott (Brandeis) | ''TBA'']]<br />
|Dymarz and Uyanik<br />
|-<br />
|April 24<br />
|Priyam Patel (Utah)<br />
|[[# Priyam Patel (Utah) | TBA ]]<br />
|Loving and Uyanik<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
===Carolyn Abbott===<br />
<br />
===Priyam Patel===<br />
<br />
== Archive of past Dynamics seminars ==<br />
<br />
2021-2022 [[Dynamics_Seminar_2021-2022]]<br />
<br />
2020-2021 [[Dynamics_Seminar_2020-2021]]</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar&diff=23335Dynamics Seminar2022-07-17T15:00:18Z<p>Cwu367: /* Fall 2022 */</p>
<hr />
<div>The [[Dynamics]] seminar meets in room '''B309 of Van Vleck Hall''' on '''Mondays''' from '''2:30pm - 3:20pm'''. To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu. For more information, contact Paul Apisa, Marissa Loving, Caglar Uyanik, or Chenxi Wu. <br />
<br />
<br />
<br />
== Fall 2022 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
<br />
|-<br />
|September 12<br />
|[https://math.ou.edu/~jing/ Jing Tao] (OU)<br />
|[[# Jing Tao (OU) | ''TBA'']]<br />
|Dymarz and Uyanik<br />
|-<br />
|September 19<br />
|[https://math.temple.edu/~tug67058/ Rebekah Palmer] (Temple)[VIRTUAL]<br />
|[[# Rebekah Palmer (Temple)| ''TBA'']]<br />
|Loving<br />
|-<br />
|September 26<br />
|TBA<br />
|[[# TBA | ''TBA'']]<br />
|<br />
<br />
|-<br />
|October 3<br />
|[https://sites.google.com/view/beibei-liu/ Beibei Liu] (MIT)<br />
|[[# Beibei Liu (Georgia Tech) | ''TBA'']]<br />
| Dymarz<br />
|-<br />
|October 10<br />
|[https://mutanguha.com/ Jean Pierre Mutanguha] (Princeton)<br />
|[[# Jean Pierre Mutanguha (Princeton) | ''TBA'']]<br />
|Uyanik<br />
|-<br />
|October 17<br />
|[https://sites.google.com/ucsd.edu/ans032/ Anthony Sanchez] (UCSD)<br />
|[[# Anthony Sanchez (UCSD)| ''TBA'']]<br />
|Uyanik<br />
|-<br />
|October 24<br />
|[https://you.stonybrook.edu/aerchenko/ Alena Erchenko] (Stony Brook)<br />
|[[# Alena Erchenko| ''TBA'']]<br />
|Uyanik and Work<br />
|-<br />
|October 31<br />
|TBA<br />
|[[# TBA| ''TBA'']]<br />
|<br />
|-<br />
|November 7<br />
|[https://sites.google.com/bc.edu/ethan-farber/about-me?authuser=0/ Ethan Farber] (BC)<br />
|[[# Ethan Farber (BC)| ''TBA'']]<br />
|Loving<br />
|-<br />
|November 14<br />
|TBA<br />
|[[# TBA| ''TBA'']]<br />
|<br />
|-<br />
|November 21<br />
|Harry Baik<br />
|[[# TBA| ''TBA'']]<br />
|Wu<br />
|-<br />
|November 28<br />
|TBA<br />
|[[# TBA| ''TBA'']]<br />
|<br />
|-<br />
|December 5<br />
|TBA<br />
|[[# TBA| ''TBA'']]<br />
|<br />
|-<br />
|December 12<br />
|TBA<br />
|[[# TBA| ''TBA'']]<br />
|<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Jing Tao===<br />
<br />
===Rebekah Palmer===<br />
<br />
===Beibei Liu===<br />
<br />
===Jean Pierre Mutanguha===<br />
<br />
===Anthony Sanchez===<br />
<br />
===Alena Erchenko===<br />
<br />
===Ethan Farber===<br />
<br />
== Spring 2023 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 30<br />
|TBA<br />
|[[TBA| ''TBA'']]<br />
|<br />
|-<br />
|March 27<br />
|[https://www.carolynrabbott.com/ Carolyn Abbott] (Brandeis)<br />
|[[# Carolyn Abbott (Brandeis) | ''TBA'']]<br />
|Dymarz and Uyanik<br />
|-<br />
|April 24<br />
|Priyam Patel (Utah)<br />
|[[# Priyam Patel (Utah) | TBA ]]<br />
|Loving and Uyanik<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
===Carolyn Abbott===<br />
<br />
===Priyam Patel===<br />
<br />
== Archive of past Dynamics seminars ==<br />
<br />
2021-2022 [[Dynamics_Seminar_2021-2022]]<br />
<br />
2020-2021 [[Dynamics_Seminar_2020-2021]]</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2021-2022&diff=21980Dynamics Seminar 2021-20222021-10-21T16:27:01Z<p>Cwu367: /* Fall 2021 */</p>
<hr />
<div>The [[Dynamics]] seminar meets in room '''901 of Van Vleck Hall''' on '''Mondays''' from '''2:30pm - 3:20pm'''. To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu. For more information, contact Caglar Uyanik or Chenxi Wu.<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
<br />
== Fall 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
<br />
|-<br />
|Sep. 13<br />
|[https://sites.tufts.edu/natefisher/ Nate Fisher] (UW Madison) <br />
|[[# Nate Fisher (UW Madison) | "Boundaries, random walks, and nilpotent groups"]]<br />
| local<br />
|-<br />
|Sep. 20<br />
|[https://www.caglaruyanik.com/ Caglar Uyanik] (UW Madison)<br />
|[[# Caglar Uyanik (UW Madison) | "Dynamics on currents and applications to free group automorphisms"]]<br />
|local<br />
|-<br />
|Sep. 27<br />
|[https://michu.people.uic.edu/ Michelle Chu] (UIC)<br />
|[[# Michelle Chu (UIC) | "Prescribed virtual torsion in the homology of 3-manifolds"]]<br />
|caglar<br />
|-<br />
|Oct. 4<br />
|[https://www.math.utah.edu/~khalil/ Osama Khalil] (Utah)<br />
|[[# Osama Khalil (Utah) | "Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation"]]<br />
|caglar<br />
|-<br />
|Oct. 11<br />
|[https://web.ma.utexas.edu/users/weisman/ Theodore Weisman] (UT Austin)<br />
|[[# Theodore Weisman (UT Austin) | "Relative Anosov representations and convex projective structures"]]<br />
|zimmer<br />
|-<br />
|Oct. 18<br />
|Grace Work (UW Madison)<br />
|[[# Grace Work (UW Madison) | "Parametrizing transversals to horocycle flow"]]<br />
|local<br />
|-<br />
|Oct. 25<br />
|Chenxi Wu (UW Madison)<br />
|[[# Chenxi Wu (UW Madison) | "Galois conjugates of exponents of quadratic core entropy"]]<br />
|local<br />
|-<br />
|Nov. 1<br />
|Jack Burkart (UW Madison)<br />
|TBA<br />
|local<br />
|-<br />
<br />
|Nov. 8<br />
|[https://faculty.washington.edu/jathreya/ Jayadev Athreya] (UW Seattle)<br />
|[[# Jayadev Athreya (UW Seattle) | "Stable Random Fields, Patterson-Sullivan Measures, and Extremal Cocycle Growth"]]<br />
|caglar and grace<br />
|-<br />
|Nov. 15<br />
|[http://math.utoledo.edu/~fgultepe/ Funda Gültepe] (U Toledo)<br />
|[[# Funda Gültepe (U Toledo) | "TBA"]]<br />
|caglar<br />
|-<br />
|Nov. 22<br />
|[https://sites.google.com/view/jonah-gaster/home Jonah Gaster] (UW Milwaukee)<br />
|[[# Jonah Gaster (UWM) | "TBA"]]<br />
|caglar<br />
|-<br />
|Nov. 29<br />
|[https://sites.google.com/view/chloe-avery/home Chloe Avery] (U Chicago)<br />
|[[# Chloe Avery (U Chicago) | "TBA"]]<br />
|Dymarz<br />
|-<br />
|Dec. 6<br />
|open<br />
|<br />
|<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Nate Fisher (UW Madison)===<br />
''Boundaries, random walks, and nilpotent groups''<br />
<br />
In this talk, we will discuss boundaries and random walks in the Heisenberg group. We will discuss a class of sub-Finsler metrics on the Heisenberg group which arise as the asymptotic cones of word metrics on the integer Heisenberg group and describe new results on the boundaries of these polygonal sub-Finsler metrics. After that, we will explore experimental work to examine the asymptotic behavior of random walks in this group. Parts of this work are joint with Sebastiano Nicolussi Golo.<br />
<br />
===Caglar Uyanik (UW Madison)===<br />
''Dynamics on currents and applications to free group automorphisms''<br />
<br />
Currents are measure theoretic generalizations of conjugacy classes on free groups, and play an important role in various low-dimensional geometry questions. I will talk about the dynamics of certain "generic" elements of Out(F) on the space of currents, and explain how it reflects on the algebraic structure of the group. <br />
<br />
<br />
===Michelle Chu (UIC)===<br />
''Prescribed virtual torsion in the homology of 3-manifolds''<br />
<br />
Hongbin Sun showed that a closed hyperbolic 3-manifold virtually contains any prescribed torsion subgroup as a direct factor in homology. In this talk we will discuss joint work with Daniel Groves generalizing Sun’s result to irreducible 3-manifolds which are not graph-manifolds.<br />
<br />
<br />
===Osama Khalil (Utah)===<br />
''Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation''<br />
<br />
Khintchine's Theorem provides a zero-one law describing the approximability of typical points by rational points. In 1984, Mahler asked whether the same holds for Cantor’s middle thirds set. His question fits into a long studied line of research aiming at showing that Diophantine sets are highly random and are thus disjoint, in a suitable sense, from highly structured sets.<br />
<br />
We will discuss the first complete analogue of Khintchine’s theorem for certain self-similar fractal measures, recently obtained in joint work with Manuel Luethi. The key ingredient in the proof is an effective equidistribution theorem for fractal measures on the space of unimodular lattices, generalizing a long history of similar results for smooth measures beginning with Sarnak’s work in the eighties. To prove the latter, we associate to such fractals certain p-adic Markov operators, reminiscent of the classical Hecke operators, and leverage their spectral properties. No background in homogeneous dynamics will be assumed.<br />
<br />
<br />
=== Theodore Weisman (UT Austin)===<br />
<br />
''Relative Anosov representations and convex projective structures''<br />
<br />
Anosov representations are a higher-rank generalization of convex cocompact subgroups of rank-one Lie groups. They are only defined for word-hyperbolic groups, but recently Kapovich-Leeb and Zhu have suggested possible definitions for an Anosov representation of a relatively hyperbolic group - aiming to give a higher-rank generalization of geometrical finiteness.<br />
<br />
In this talk, we will introduce a more general version of relative Anosov representation which also interacts well with the theory of convex projective structures. In particular, the definition includes projectively convex cocompact representations of relatively hyperbolic groups, and allows for deformations of cusped convex projective manifolds (including hyperbolic manifolds) in which the cusp groups change in nontrivial ways.<br />
<br />
===Grace Work (UW Madison)===<br />
There are many interesting dynamical flows that arise in the context of translation surfaces, including the horocycle flow. One application of the horocycle flow is to compute the distribution of the gaps between slopes of saddle connections on a specific translation surface. This method was first developed by Athreya and Chueng in the case of the torus, where the question can be restated in terms of Farey fractions and was solved by R. R. Hall using methods from analytic number theory. An important step in this process is to find a good parametrization of a transversal to horocycle flow. We will show how to do this explicitly in the case of the octagon, how it generalizes to a specific class of translation surfaces, lattice surfaces, (both joint work with Caglar Uyanik), and examine how to parametrize the transversal for a generic surface in a given moduli space.<br />
<br />
===Chenxi Wu (UW Madison)===<br />
The Hubbard tree is a combinatorial object that encodes the dynamic of a post critically finite polynomial map, and its topological entropy is called the core entropy. I will talk about an upcoming paper with Kathryn Lindsey and Giulio Tiozzo where we provide geometric constrains to the Galois conjugates of exponents of core entropy, which gives a necessary condition for a number to be the core entropy for a super attracting parameter.<br />
<br />
===Jack Burkart (UW Madison)===<br />
"TBA"<br />
<br />
===Jayadev Athreya (UW Seattle)===<br />
''Stable Random Fields, Patterson-Sullivan Measures, and Extremal Cocycle Growth''<br />
<br />
We study extreme values of group-indexed stable random fields<br />
for discrete groups G acting geometrically on spaces X in the following cases:<br />
(1) G acts freely, properly discontinuously by isometries on a CAT(-1) space X,<br />
(2) G is a lattice in a higher rank Lie group, acting on a symmetric space X,<br />
(3) G is the mapping class group of a surface acting on its Teichmuller space. The connection between extreme values and the geometric action is mediated by the action of the group G on its limit set equipped with the Patterson-Sullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth which measures the distortion of measures on the boundary in comparison to the movement of points in the space X and show that its non-vanishing is equivalent to<br />
finiteness of the Bowen-Margulis measure for the associated unit tangent bundle U(X/G) provided X/G has non-arithmetic length spectrum. This is joint work with Mahan MJ and Parthanil Roy.<br />
<br />
===Funda Gültepe (U Toledo)===<br />
"TBA"<br />
<br />
===Jonah Gaster (UWM)===<br />
''TBA''<br />
<br />
===Chloe Avery (U Chicago)===<br />
"TBA"<br />
<br />
== Archive of past Dynamics seminars ==<br />
<br />
2020-2021 [[Dynamics_Seminar_2020-2021]]</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2021-2022&diff=21979Dynamics Seminar 2021-20222021-10-21T16:26:33Z<p>Cwu367: /* Abstracts */</p>
<hr />
<div>The [[Dynamics]] seminar meets in room '''901 of Van Vleck Hall''' on '''Mondays''' from '''2:30pm - 3:20pm'''. To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu. For more information, contact Caglar Uyanik or Chenxi Wu.<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
<br />
== Fall 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
<br />
|-<br />
|Sep. 13<br />
|[https://sites.tufts.edu/natefisher/ Nate Fisher] (UW Madison) <br />
|[[# Nate Fisher (UW Madison) | "Boundaries, random walks, and nilpotent groups"]]<br />
| local<br />
|-<br />
|Sep. 20<br />
|[https://www.caglaruyanik.com/ Caglar Uyanik] (UW Madison)<br />
|[[# Caglar Uyanik (UW Madison) | "Dynamics on currents and applications to free group automorphisms"]]<br />
|local<br />
|-<br />
|Sep. 27<br />
|[https://michu.people.uic.edu/ Michelle Chu] (UIC)<br />
|[[# Michelle Chu (UIC) | "Prescribed virtual torsion in the homology of 3-manifolds"]]<br />
|caglar<br />
|-<br />
|Oct. 4<br />
|[https://www.math.utah.edu/~khalil/ Osama Khalil] (Utah)<br />
|[[# Osama Khalil (Utah) | "Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation"]]<br />
|caglar<br />
|-<br />
|Oct. 11<br />
|[https://web.ma.utexas.edu/users/weisman/ Theodore Weisman] (UT Austin)<br />
|[[# Theodore Weisman (UT Austin) | "Relative Anosov representations and convex projective structures"]]<br />
|zimmer<br />
|-<br />
|Oct. 18<br />
|Grace Work (UW Madison)<br />
|[[# Grace Work (UW Madison) | "Parametrizing transversals to horocycle flow"]]<br />
|local<br />
|-<br />
|Oct. 25<br />
|Chenxi Wu (UW Madison)<br />
|[[# Chenxi Wu (UW Madison) | "Galois conjugates of quadratic core entropy"]]<br />
|local<br />
|-<br />
|Nov. 1<br />
|Jack Burkart (UW Madison)<br />
|TBA<br />
|local<br />
|-<br />
<br />
|Nov. 8<br />
|[https://faculty.washington.edu/jathreya/ Jayadev Athreya] (UW Seattle)<br />
|[[# Jayadev Athreya (UW Seattle) | "Stable Random Fields, Patterson-Sullivan Measures, and Extremal Cocycle Growth"]]<br />
|caglar and grace<br />
|-<br />
|Nov. 15<br />
|[http://math.utoledo.edu/~fgultepe/ Funda Gültepe] (U Toledo)<br />
|[[# Funda Gültepe (U Toledo) | "TBA"]]<br />
|caglar<br />
|-<br />
|Nov. 22<br />
|[https://sites.google.com/view/jonah-gaster/home Jonah Gaster] (UW Milwaukee)<br />
|[[# Jonah Gaster (UWM) | "TBA"]]<br />
|caglar<br />
|-<br />
|Nov. 29<br />
|[https://sites.google.com/view/chloe-avery/home Chloe Avery] (U Chicago)<br />
|[[# Chloe Avery (U Chicago) | "TBA"]]<br />
|Dymarz<br />
|-<br />
|Dec. 6<br />
|open<br />
|<br />
|<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Nate Fisher (UW Madison)===<br />
''Boundaries, random walks, and nilpotent groups''<br />
<br />
In this talk, we will discuss boundaries and random walks in the Heisenberg group. We will discuss a class of sub-Finsler metrics on the Heisenberg group which arise as the asymptotic cones of word metrics on the integer Heisenberg group and describe new results on the boundaries of these polygonal sub-Finsler metrics. After that, we will explore experimental work to examine the asymptotic behavior of random walks in this group. Parts of this work are joint with Sebastiano Nicolussi Golo.<br />
<br />
===Caglar Uyanik (UW Madison)===<br />
''Dynamics on currents and applications to free group automorphisms''<br />
<br />
Currents are measure theoretic generalizations of conjugacy classes on free groups, and play an important role in various low-dimensional geometry questions. I will talk about the dynamics of certain "generic" elements of Out(F) on the space of currents, and explain how it reflects on the algebraic structure of the group. <br />
<br />
<br />
===Michelle Chu (UIC)===<br />
''Prescribed virtual torsion in the homology of 3-manifolds''<br />
<br />
Hongbin Sun showed that a closed hyperbolic 3-manifold virtually contains any prescribed torsion subgroup as a direct factor in homology. In this talk we will discuss joint work with Daniel Groves generalizing Sun’s result to irreducible 3-manifolds which are not graph-manifolds.<br />
<br />
<br />
===Osama Khalil (Utah)===<br />
''Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation''<br />
<br />
Khintchine's Theorem provides a zero-one law describing the approximability of typical points by rational points. In 1984, Mahler asked whether the same holds for Cantor’s middle thirds set. His question fits into a long studied line of research aiming at showing that Diophantine sets are highly random and are thus disjoint, in a suitable sense, from highly structured sets.<br />
<br />
We will discuss the first complete analogue of Khintchine’s theorem for certain self-similar fractal measures, recently obtained in joint work with Manuel Luethi. The key ingredient in the proof is an effective equidistribution theorem for fractal measures on the space of unimodular lattices, generalizing a long history of similar results for smooth measures beginning with Sarnak’s work in the eighties. To prove the latter, we associate to such fractals certain p-adic Markov operators, reminiscent of the classical Hecke operators, and leverage their spectral properties. No background in homogeneous dynamics will be assumed.<br />
<br />
<br />
=== Theodore Weisman (UT Austin)===<br />
<br />
''Relative Anosov representations and convex projective structures''<br />
<br />
Anosov representations are a higher-rank generalization of convex cocompact subgroups of rank-one Lie groups. They are only defined for word-hyperbolic groups, but recently Kapovich-Leeb and Zhu have suggested possible definitions for an Anosov representation of a relatively hyperbolic group - aiming to give a higher-rank generalization of geometrical finiteness.<br />
<br />
In this talk, we will introduce a more general version of relative Anosov representation which also interacts well with the theory of convex projective structures. In particular, the definition includes projectively convex cocompact representations of relatively hyperbolic groups, and allows for deformations of cusped convex projective manifolds (including hyperbolic manifolds) in which the cusp groups change in nontrivial ways.<br />
<br />
===Grace Work (UW Madison)===<br />
There are many interesting dynamical flows that arise in the context of translation surfaces, including the horocycle flow. One application of the horocycle flow is to compute the distribution of the gaps between slopes of saddle connections on a specific translation surface. This method was first developed by Athreya and Chueng in the case of the torus, where the question can be restated in terms of Farey fractions and was solved by R. R. Hall using methods from analytic number theory. An important step in this process is to find a good parametrization of a transversal to horocycle flow. We will show how to do this explicitly in the case of the octagon, how it generalizes to a specific class of translation surfaces, lattice surfaces, (both joint work with Caglar Uyanik), and examine how to parametrize the transversal for a generic surface in a given moduli space.<br />
<br />
===Chenxi Wu (UW Madison)===<br />
The Hubbard tree is a combinatorial object that encodes the dynamic of a post critically finite polynomial map, and its topological entropy is called the core entropy. I will talk about an upcoming paper with Kathryn Lindsey and Giulio Tiozzo where we provide geometric constrains to the Galois conjugates of exponents of core entropy, which gives a necessary condition for a number to be the core entropy for a super attracting parameter.<br />
<br />
===Jack Burkart (UW Madison)===<br />
"TBA"<br />
<br />
===Jayadev Athreya (UW Seattle)===<br />
''Stable Random Fields, Patterson-Sullivan Measures, and Extremal Cocycle Growth''<br />
<br />
We study extreme values of group-indexed stable random fields<br />
for discrete groups G acting geometrically on spaces X in the following cases:<br />
(1) G acts freely, properly discontinuously by isometries on a CAT(-1) space X,<br />
(2) G is a lattice in a higher rank Lie group, acting on a symmetric space X,<br />
(3) G is the mapping class group of a surface acting on its Teichmuller space. The connection between extreme values and the geometric action is mediated by the action of the group G on its limit set equipped with the Patterson-Sullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth which measures the distortion of measures on the boundary in comparison to the movement of points in the space X and show that its non-vanishing is equivalent to<br />
finiteness of the Bowen-Margulis measure for the associated unit tangent bundle U(X/G) provided X/G has non-arithmetic length spectrum. This is joint work with Mahan MJ and Parthanil Roy.<br />
<br />
===Funda Gültepe (U Toledo)===<br />
"TBA"<br />
<br />
===Jonah Gaster (UWM)===<br />
''TBA''<br />
<br />
===Chloe Avery (U Chicago)===<br />
"TBA"<br />
<br />
== Archive of past Dynamics seminars ==<br />
<br />
2020-2021 [[Dynamics_Seminar_2020-2021]]</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2021-2022&diff=21978Dynamics Seminar 2021-20222021-10-21T16:18:23Z<p>Cwu367: /* Fall 2021 */</p>
<hr />
<div>The [[Dynamics]] seminar meets in room '''901 of Van Vleck Hall''' on '''Mondays''' from '''2:30pm - 3:20pm'''. To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu. For more information, contact Caglar Uyanik or Chenxi Wu.<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
<br />
== Fall 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
<br />
|-<br />
|Sep. 13<br />
|[https://sites.tufts.edu/natefisher/ Nate Fisher] (UW Madison) <br />
|[[# Nate Fisher (UW Madison) | "Boundaries, random walks, and nilpotent groups"]]<br />
| local<br />
|-<br />
|Sep. 20<br />
|[https://www.caglaruyanik.com/ Caglar Uyanik] (UW Madison)<br />
|[[# Caglar Uyanik (UW Madison) | "Dynamics on currents and applications to free group automorphisms"]]<br />
|local<br />
|-<br />
|Sep. 27<br />
|[https://michu.people.uic.edu/ Michelle Chu] (UIC)<br />
|[[# Michelle Chu (UIC) | "Prescribed virtual torsion in the homology of 3-manifolds"]]<br />
|caglar<br />
|-<br />
|Oct. 4<br />
|[https://www.math.utah.edu/~khalil/ Osama Khalil] (Utah)<br />
|[[# Osama Khalil (Utah) | "Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation"]]<br />
|caglar<br />
|-<br />
|Oct. 11<br />
|[https://web.ma.utexas.edu/users/weisman/ Theodore Weisman] (UT Austin)<br />
|[[# Theodore Weisman (UT Austin) | "Relative Anosov representations and convex projective structures"]]<br />
|zimmer<br />
|-<br />
|Oct. 18<br />
|Grace Work (UW Madison)<br />
|[[# Grace Work (UW Madison) | "Parametrizing transversals to horocycle flow"]]<br />
|local<br />
|-<br />
|Oct. 25<br />
|Chenxi Wu (UW Madison)<br />
|[[# Chenxi Wu (UW Madison) | "Galois conjugates of quadratic core entropy"]]<br />
|local<br />
|-<br />
|Nov. 1<br />
|Jack Burkart (UW Madison)<br />
|TBA<br />
|local<br />
|-<br />
<br />
|Nov. 8<br />
|[https://faculty.washington.edu/jathreya/ Jayadev Athreya] (UW Seattle)<br />
|[[# Jayadev Athreya (UW Seattle) | "Stable Random Fields, Patterson-Sullivan Measures, and Extremal Cocycle Growth"]]<br />
|caglar and grace<br />
|-<br />
|Nov. 15<br />
|[http://math.utoledo.edu/~fgultepe/ Funda Gültepe] (U Toledo)<br />
|[[# Funda Gültepe (U Toledo) | "TBA"]]<br />
|caglar<br />
|-<br />
|Nov. 22<br />
|[https://sites.google.com/view/jonah-gaster/home Jonah Gaster] (UW Milwaukee)<br />
|[[# Jonah Gaster (UWM) | "TBA"]]<br />
|caglar<br />
|-<br />
|Nov. 29<br />
|[https://sites.google.com/view/chloe-avery/home Chloe Avery] (U Chicago)<br />
|[[# Chloe Avery (U Chicago) | "TBA"]]<br />
|Dymarz<br />
|-<br />
|Dec. 6<br />
|open<br />
|<br />
|<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Nate Fisher (UW Madison)===<br />
''Boundaries, random walks, and nilpotent groups''<br />
<br />
In this talk, we will discuss boundaries and random walks in the Heisenberg group. We will discuss a class of sub-Finsler metrics on the Heisenberg group which arise as the asymptotic cones of word metrics on the integer Heisenberg group and describe new results on the boundaries of these polygonal sub-Finsler metrics. After that, we will explore experimental work to examine the asymptotic behavior of random walks in this group. Parts of this work are joint with Sebastiano Nicolussi Golo.<br />
<br />
===Caglar Uyanik (UW Madison)===<br />
''Dynamics on currents and applications to free group automorphisms''<br />
<br />
Currents are measure theoretic generalizations of conjugacy classes on free groups, and play an important role in various low-dimensional geometry questions. I will talk about the dynamics of certain "generic" elements of Out(F) on the space of currents, and explain how it reflects on the algebraic structure of the group. <br />
<br />
<br />
===Michelle Chu (UIC)===<br />
''Prescribed virtual torsion in the homology of 3-manifolds''<br />
<br />
Hongbin Sun showed that a closed hyperbolic 3-manifold virtually contains any prescribed torsion subgroup as a direct factor in homology. In this talk we will discuss joint work with Daniel Groves generalizing Sun’s result to irreducible 3-manifolds which are not graph-manifolds.<br />
<br />
<br />
===Osama Khalil (Utah)===<br />
''Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation''<br />
<br />
Khintchine's Theorem provides a zero-one law describing the approximability of typical points by rational points. In 1984, Mahler asked whether the same holds for Cantor’s middle thirds set. His question fits into a long studied line of research aiming at showing that Diophantine sets are highly random and are thus disjoint, in a suitable sense, from highly structured sets.<br />
<br />
We will discuss the first complete analogue of Khintchine’s theorem for certain self-similar fractal measures, recently obtained in joint work with Manuel Luethi. The key ingredient in the proof is an effective equidistribution theorem for fractal measures on the space of unimodular lattices, generalizing a long history of similar results for smooth measures beginning with Sarnak’s work in the eighties. To prove the latter, we associate to such fractals certain p-adic Markov operators, reminiscent of the classical Hecke operators, and leverage their spectral properties. No background in homogeneous dynamics will be assumed.<br />
<br />
<br />
=== Theodore Weisman (UT Austin)===<br />
<br />
''Relative Anosov representations and convex projective structures''<br />
<br />
Anosov representations are a higher-rank generalization of convex cocompact subgroups of rank-one Lie groups. They are only defined for word-hyperbolic groups, but recently Kapovich-Leeb and Zhu have suggested possible definitions for an Anosov representation of a relatively hyperbolic group - aiming to give a higher-rank generalization of geometrical finiteness.<br />
<br />
In this talk, we will introduce a more general version of relative Anosov representation which also interacts well with the theory of convex projective structures. In particular, the definition includes projectively convex cocompact representations of relatively hyperbolic groups, and allows for deformations of cusped convex projective manifolds (including hyperbolic manifolds) in which the cusp groups change in nontrivial ways.<br />
<br />
===Grace Work (UW Madison)===<br />
There are many interesting dynamical flows that arise in the context of translation surfaces, including the horocycle flow. One application of the horocycle flow is to compute the distribution of the gaps between slopes of saddle connections on a specific translation surface. This method was first developed by Athreya and Chueng in the case of the torus, where the question can be restated in terms of Farey fractions and was solved by R. R. Hall using methods from analytic number theory. An important step in this process is to find a good parametrization of a transversal to horocycle flow. We will show how to do this explicitly in the case of the octagon, how it generalizes to a specific class of translation surfaces, lattice surfaces, (both joint work with Caglar Uyanik), and examine how to parametrize the transversal for a generic surface in a given moduli space.<br />
<br />
===Chenxi Wu (UW Madison)===<br />
"TBA"<br />
<br />
===Jack Burkart (UW Madison)===<br />
"TBA"<br />
<br />
===Jayadev Athreya (UW Seattle)===<br />
''Stable Random Fields, Patterson-Sullivan Measures, and Extremal Cocycle Growth''<br />
<br />
We study extreme values of group-indexed stable random fields<br />
for discrete groups G acting geometrically on spaces X in the following cases:<br />
(1) G acts freely, properly discontinuously by isometries on a CAT(-1) space X,<br />
(2) G is a lattice in a higher rank Lie group, acting on a symmetric space X,<br />
(3) G is the mapping class group of a surface acting on its Teichmuller space. The connection between extreme values and the geometric action is mediated by the action of the group G on its limit set equipped with the Patterson-Sullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth which measures the distortion of measures on the boundary in comparison to the movement of points in the space X and show that its non-vanishing is equivalent to<br />
finiteness of the Bowen-Margulis measure for the associated unit tangent bundle U(X/G) provided X/G has non-arithmetic length spectrum. This is joint work with Mahan MJ and Parthanil Roy.<br />
<br />
===Funda Gültepe (U Toledo)===<br />
"TBA"<br />
<br />
===Jonah Gaster (UWM)===<br />
''TBA''<br />
<br />
===Chloe Avery (U Chicago)===<br />
"TBA"<br />
<br />
<br />
<br />
<br />
<br />
== Archive of past Dynamics seminars ==<br />
<br />
2020-2021 [[Dynamics_Seminar_2020-2021]]</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=21180Dynamics Seminar 2020-20212021-04-27T17:23:59Z<p>Cwu367: </p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
Meetings are on Zoom. To get Zoom info email Chenxi Wu. <br />
<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
== Spring 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|February 3<br />
|Daniel Woodhouse (Oxford)<br />
|Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups<br />
| <br />
|-<br />
|February 10<br />
|John Mackay (Bristol)<br />
|Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy<br />
| <br />
|-<br />
|February 17<br />
|Benjamin Branman (Wisconsin)<br />
|Spaces of Pants Decompositions for Surfaces of Infinite Type<br />
| <br />
|-<br />
|February 24<br />
|Uri Bader (Weizmann Institute)<br />
|Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity.<br />
| <br />
|-<br />
|March 3<br />
|Omri Sarig (Weizmann Institute)<br />
|(Dis)continuity of Lyapunov exponents for surface diffeomorphisms (joint with J. Buzz and S. Crovisier)<br />
| <br />
|-<br />
|March 10<br />
|Chris Leininger (Rice University)<br />
|Billiards, symbolic coding, and cone metrics<br />
| <br />
|-<br />
|March 17<br />
|Ethan Farber (Boston College)<br />
|Constructing pseudo-Anosovs from expanding interval maps<br />
|<br />
|-<br />
|March 24<br />
|Jon Chaika (Utah)<br />
|A strange limit of horocycle ergodic measures in a stratum of<br />
translation surfaces<br />
|<br />
|-<br />
|March 31<br />
|Harrison Bray (George Mason)<br />
|Volume-entropy rigidity for convex real projective manifolds<br />
|<br />
|-<br />
|April 7<br />
|Claire Burrin （ETH Zurich)<br />
|A sparse equidistribution problem for expanding horocycles on the modular surface<br />
|<br />
|-<br />
|April 21<br />
|Kasra Rafi (Toronto)<br />
|Absolutely continuous stationary measures for the mapping class group<br />
|<br />
|-<br />
|April 28<br />
|Matt Bainbridge (Indiana)<br />
|Haupt's theorem for strata of holomorphic one-forms and isoperiodic foliations<br />
|}<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
===Daniel Woodhouse===<br />
<br />
"Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups"<br />
<br />
Let F be a finitely rank free group.<br />
Let w_1 and w_2 be suitable random/generic elements in F.<br />
Consider the HNN extension G = <F, t | t w_1 t^{-1} = w_2 >.<br />
It is known from existing results that G will be 1-ended and hyperbolic.<br />
We have shown that G is quasi-isometrically rigid.<br />
That is to say that if a f.g. group H is quasi-isometric to G, then G and H are virtually isomorphic.<br />
The full result is for finite graphs of groups with virtually free vertex groups and and two-ended edge groups, but the statement is more technical -- not all such groups are QI-rigid.<br />
The main argument involves applying a new proof of Leighton's graph covering theorem.<br />
This is joint work with Sam Shepherd.<br />
<br />
===John Mackay===<br />
<br />
"Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy"<br />
<br />
The separation profile of an infinite graph was introduced by<br />
Benjamini-Schramm-Timar. It is a function which measures how<br />
well-connected the graph is by how hard it is to cut finite subgraphs<br />
into small pieces. In earlier joint work with David Hume and Romain<br />
Tessera, we introduced Poincaré profiles, generalising this concept by<br />
using p-Poincaré inequalities to measure the connected-ness of<br />
subgraphs. I will discuss this family of invariants, their applications<br />
to coarse embedding problems, and recent work finding the profiles of<br />
all connected unimodular Lie groups, where a dichotomy is exhibited.<br />
Joint with Hume and Tessera.<br />
<br />
===Benjamin Branman===<br />
<br />
"Spaces of Pants Decompositions for Surfaces of Infinite Type"<br />
<br />
We study the pants graph of surfaces of infinite type. When S is a surface of infinite type, the usual definition of the graph of pants decompositions yields a graph with infinitely many connected-components. In the first part of our talk, we study this disconnected graph. In particular, we show that the extended mapping class group of S is isomorphic to a proper subgroup of of the pants graph, in contrast to the finite-type case. In the second part of the talk, motivated by the Metaconjecture of Ivanov, we seek to endow the pants graph with additional structure. To this end, we define a coarser topology on the pants graph than the topology inherited from the graph structure. We show that our new space is path-connected, and that its automorphism group is isomorphic to the extended mapping class group.<br />
<br />
<br />
===Uri Bader===<br />
"Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity."<br />
<br />
Compact hyperbolic manifolds are very interesting geometric objects.<br />
Maybe surprisingly, they are also interesting from an algebraic point of view:<br />
They are completely determined by their fundamental groups (this is Mostow's Theorem),<br />
which could be seen as a subgroup of the integer valued invertible matrices in some dimension, GL_n(Z).<br />
When the fundamental group is the Z-points of some algebraic subgroup of GL_n we say that the manifold is arithmetic.<br />
A question arises: is there a simple geometric criterion for arithmeticity for hyperbolic manifolds?<br />
Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen.<br />
In a recent work with Fisher, Miller and Stover we proved this conjecture.<br />
Our proof is based on the theory of AREA, namely Algebraic Representation of Ergodic Actions, which I have developed with Alex Furman in recent years.<br />
In this talk I will try to survey the subject in a colloquial manner.<br />
<br />
<br />
===Omri Sarig===<br />
<br />
"(Dis)continuity of Lyapunov exponents for surface diffeomorphisms" (joint with J. Buzz and S. Crovisier)"<br />
<br />
Let f be an infinitely differentiable surface diffeomorphism. Suppose we are given a sequence of ergodic invariant measures m_n which converge weak star to an ergodic limit m. What do we need to know on m_n to guarantee that the Lyapunov exponents of m_n converge to the Lyapunov exponents of m?<br />
The main result is that if m has positive entropy, and the entropy of m_n converges to the entropy of m, then the Lyapunov exponents of m_n converge to the Lyapunov exponents of m.<br />
This is joint work with J. Buzzi and S. Crovisier.<br />
<br />
===Chris Leininger===<br />
<br />
"Billiards, symbolic coding, and cone metrics"<br />
<br />
Given a polygon in the Euclidean or hyperbolic plane a billiard trajectory in the polygon is the geodesic path of a particle in the polygon bouncing off the sides so that the angle of reflection is equal to the angle incidence. A billiard trajectory determines a symbolic coding via the sides of the polygon encountered. In this talk I will describe joint work with Erlandsson and Sadanand showing the extent to which the set of all coding sequences, the bounce spectrum, determines the shape of a hyperbolic polygon. We completely characterize those polygons which are billiard rigid (the generic case), meaning that they are determined up to isometry by their bounce spectrum. When rigidity fails for a polygon P, we parameterize the space of polygons having the same bounce spectrum at P. These results for billiards are a consequence of a rigidity/flexibility theorem for negatively curved hyperbolic cone metrics. In the talk I will explain the theorem about hyperbolic billiards, comparing/contrasting it with the Euclidean case (earlier work with Duchin, Erlandsson, and Sadanand). Then I will explain the relationship with hyperbolic cone metrics, state our rigidity/flexibility theorem for such metrics, and as time allows describe some of the ideas involved in the proofs.<br />
<br />
===Ethan Farber===<br />
<br />
"Constructing pseudo-Anosovs from expanding interval maps"<br />
<br />
The celebrated Nielsen-Thurston classification of surface homeomorphisms says that, up to isotopy, there are three types of homeomorphisms of a closed, connected surface: (1) finite order, (2) reducible, and (3) pseudo-Anosov. Of these three types, pseudo-Anosovs are the most intriguing to dynamicists, with connections to symbolic dynamics and flat geometry. In this talk we investigate a construction of generalized pseudo-Anosovs from interval maps, first introduced by de Carvalho. In particular, for a certain class of interval maps we give necessary and sufficient conditions for the construction to produce a true pseudo-Anosov, which may be recast in terms of the kneading data of the interval map. We also describe a bijection between such interval maps and the rationals in the open unit interval which captures the kneading data, and which increases monotonically in the entropy of the interval map.<br />
<br />
===Jon Chaika===<br />
<br />
"A strange limit of horocycle ergodic measures in a stratum of<br />
translation surfaces"<br />
<br />
The main result of this talk is that in the space of unit area<br />
translation surfaces with one cone point there is a weak-star limit of<br />
measures on periodic horocycles that is fully supported in the<br />
7-dimensional space but gives positive measure to a 3-dimensional<br />
submanifold. As a consequence we obtain a non-genericity result for the<br />
horocycle flow in this space. I will define the terminology. This is joint<br />
work with Osama Khalil and John Smillie.<br />
<br />
===Harrison Bray===<br />
<br />
"Volume-entropy rigidity for convex real projective manifolds"<br />
<br />
I will discuss joint work with Constantine, building on joint work with Adeboye and Constantine, on a volume-entropy rigidity result for finite volume strictly convex projective manifolds in dimension at least 3. The result is a Besson-Courtois-Gallot type theorem, using the barycenter method. As an application, we get a uniform lower bound on the Hilbert volume of a finite volume strictly convex projective manifold of dimension at least 3.<br />
<br />
===Claire Burrin===<br />
<br />
"A sparse equidistribution problem for expanding horocycles on the modular surface"<br />
<br />
Abstract: The orbits of the horocycle flow on hyperbolic surfaces (or orbifolds) are classified: each orbit is either dense or a closed horocycle around a cusp. Expanding closed horocycles are themselves asymptotically dense, and in fact become equidistributed on the surface. The precise rate of equidistribution is of interest; on the modular surface, Zagier observed that a particular rate is equivalent to the Riemann hypothesis being true. In this talk, I will discuss the asymptotic behavior of evenly spaced points along an expanding closed horocycle on the modular surface. In this problem, the number of points depends on the expansion rate of the horocycle, and the difficulty is that these points are no more invariant under the horocycle flow. This is based on joint work with Uri Shapira and Shucheng Yu.<br />
<br />
===Kasra Rafi===<br />
<br />
"Absolutely continuous stationary measures for the mapping class group"<br />
<br />
We prove a version of a Theorem of Furstenberg in the setting of Mapping class groups. Thurston measure defines a smooth measure class on space of projectivized measured laminations For every measure \nu in this measure class, we produce a measure \mu with finite first moment on the mapping class group such that \nu is the unique \mu-stationary measure. In particular, this gives an coding-free proof of the already known result that the Lyapunov spectrum of Kontsevich-Zorich cocycle on the principal stratum of quadratic differentials is simple. This is a joint work with Alex Eskin and Maryam Mirzakhani. <br />
<br />
===Matt Bainbridge===<br />
<br />
"Haupt's theorem for strata of holomorphic one-forms and isoperiodic foliations"<br />
<br />
Haupt's Theorem (dating back to 1920) characterizes the cohomology classes of the holomorphic one-forms on a surface S with respect to any complex structure on S. More recently, Haupt's theorem was rediscovered by Kapovich, who gave a dynamical proof via Ratner's Theorem. In this talk, I'll give a refinement of Haupt's theorem characterizing the cohomology classes of holomorphic one-forms which have zeros of specified orders. The proof uses recent work of Calsamiglia, Deroin, and Francaviglia on the dynamics of the isoperiodic foliation of the moduli space of holomorphic one-forms. This is joint work with Chris Johnson, Chris Judge, and InSung Park.<br />
<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=21160Dynamics Seminar 2020-20212021-04-20T18:09:53Z<p>Cwu367: </p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
Meetings are on Zoom. To get Zoom info email Chenxi Wu. <br />
<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
== Spring 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|February 3<br />
|Daniel Woodhouse (Oxford)<br />
|Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups<br />
| <br />
|-<br />
|February 10<br />
|John Mackay (Bristol)<br />
|Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy<br />
| <br />
|-<br />
|February 17<br />
|Benjamin Branman (Wisconsin)<br />
|Spaces of Pants Decompositions for Surfaces of Infinite Type<br />
| <br />
|-<br />
|February 24<br />
|Uri Bader (Weizmann Institute)<br />
|Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity.<br />
| <br />
|-<br />
|March 3<br />
|Omri Sarig (Weizmann Institute)<br />
|(Dis)continuity of Lyapunov exponents for surface diffeomorphisms (joint with J. Buzz and S. Crovisier)<br />
| <br />
|-<br />
|March 10<br />
|Chris Leininger (Rice University)<br />
|Billiards, symbolic coding, and cone metrics<br />
| <br />
|-<br />
|March 17<br />
|Ethan Farber (Boston College)<br />
|Constructing pseudo-Anosovs from expanding interval maps<br />
|<br />
|-<br />
|March 24<br />
|Jon Chaika (Utah)<br />
|A strange limit of horocycle ergodic measures in a stratum of<br />
translation surfaces<br />
|<br />
|-<br />
|March 31<br />
|Harrison Bray (George Mason)<br />
|Volume-entropy rigidity for convex real projective manifolds<br />
|<br />
|-<br />
|April 7<br />
|Claire Burrin （ETH Zurich)<br />
|A sparse equidistribution problem for expanding horocycles on the modular surface<br />
|<br />
|-<br />
|April 21<br />
|Kasra Rafi (Toronto)<br />
|Absolutely continuous stationary measures for the mapping class group<br />
|<br />
|-<br />
|April 28<br />
|Matt Bainbridge (Indiana)<br />
|TBA<br />
|}<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
===Daniel Woodhouse===<br />
<br />
"Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups"<br />
<br />
Let F be a finitely rank free group.<br />
Let w_1 and w_2 be suitable random/generic elements in F.<br />
Consider the HNN extension G = <F, t | t w_1 t^{-1} = w_2 >.<br />
It is known from existing results that G will be 1-ended and hyperbolic.<br />
We have shown that G is quasi-isometrically rigid.<br />
That is to say that if a f.g. group H is quasi-isometric to G, then G and H are virtually isomorphic.<br />
The full result is for finite graphs of groups with virtually free vertex groups and and two-ended edge groups, but the statement is more technical -- not all such groups are QI-rigid.<br />
The main argument involves applying a new proof of Leighton's graph covering theorem.<br />
This is joint work with Sam Shepherd.<br />
<br />
===John Mackay===<br />
<br />
"Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy"<br />
<br />
The separation profile of an infinite graph was introduced by<br />
Benjamini-Schramm-Timar. It is a function which measures how<br />
well-connected the graph is by how hard it is to cut finite subgraphs<br />
into small pieces. In earlier joint work with David Hume and Romain<br />
Tessera, we introduced Poincaré profiles, generalising this concept by<br />
using p-Poincaré inequalities to measure the connected-ness of<br />
subgraphs. I will discuss this family of invariants, their applications<br />
to coarse embedding problems, and recent work finding the profiles of<br />
all connected unimodular Lie groups, where a dichotomy is exhibited.<br />
Joint with Hume and Tessera.<br />
<br />
===Benjamin Branman===<br />
<br />
"Spaces of Pants Decompositions for Surfaces of Infinite Type"<br />
<br />
We study the pants graph of surfaces of infinite type. When S is a surface of infinite type, the usual definition of the graph of pants decompositions yields a graph with infinitely many connected-components. In the first part of our talk, we study this disconnected graph. In particular, we show that the extended mapping class group of S is isomorphic to a proper subgroup of of the pants graph, in contrast to the finite-type case. In the second part of the talk, motivated by the Metaconjecture of Ivanov, we seek to endow the pants graph with additional structure. To this end, we define a coarser topology on the pants graph than the topology inherited from the graph structure. We show that our new space is path-connected, and that its automorphism group is isomorphic to the extended mapping class group.<br />
<br />
<br />
===Uri Bader===<br />
"Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity."<br />
<br />
Compact hyperbolic manifolds are very interesting geometric objects.<br />
Maybe surprisingly, they are also interesting from an algebraic point of view:<br />
They are completely determined by their fundamental groups (this is Mostow's Theorem),<br />
which could be seen as a subgroup of the integer valued invertible matrices in some dimension, GL_n(Z).<br />
When the fundamental group is the Z-points of some algebraic subgroup of GL_n we say that the manifold is arithmetic.<br />
A question arises: is there a simple geometric criterion for arithmeticity for hyperbolic manifolds?<br />
Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen.<br />
In a recent work with Fisher, Miller and Stover we proved this conjecture.<br />
Our proof is based on the theory of AREA, namely Algebraic Representation of Ergodic Actions, which I have developed with Alex Furman in recent years.<br />
In this talk I will try to survey the subject in a colloquial manner.<br />
<br />
<br />
===Omri Sarig===<br />
<br />
"(Dis)continuity of Lyapunov exponents for surface diffeomorphisms" (joint with J. Buzz and S. Crovisier)"<br />
<br />
Let f be an infinitely differentiable surface diffeomorphism. Suppose we are given a sequence of ergodic invariant measures m_n which converge weak star to an ergodic limit m. What do we need to know on m_n to guarantee that the Lyapunov exponents of m_n converge to the Lyapunov exponents of m?<br />
The main result is that if m has positive entropy, and the entropy of m_n converges to the entropy of m, then the Lyapunov exponents of m_n converge to the Lyapunov exponents of m.<br />
This is joint work with J. Buzzi and S. Crovisier.<br />
<br />
===Chris Leininger===<br />
<br />
"Billiards, symbolic coding, and cone metrics"<br />
<br />
Given a polygon in the Euclidean or hyperbolic plane a billiard trajectory in the polygon is the geodesic path of a particle in the polygon bouncing off the sides so that the angle of reflection is equal to the angle incidence. A billiard trajectory determines a symbolic coding via the sides of the polygon encountered. In this talk I will describe joint work with Erlandsson and Sadanand showing the extent to which the set of all coding sequences, the bounce spectrum, determines the shape of a hyperbolic polygon. We completely characterize those polygons which are billiard rigid (the generic case), meaning that they are determined up to isometry by their bounce spectrum. When rigidity fails for a polygon P, we parameterize the space of polygons having the same bounce spectrum at P. These results for billiards are a consequence of a rigidity/flexibility theorem for negatively curved hyperbolic cone metrics. In the talk I will explain the theorem about hyperbolic billiards, comparing/contrasting it with the Euclidean case (earlier work with Duchin, Erlandsson, and Sadanand). Then I will explain the relationship with hyperbolic cone metrics, state our rigidity/flexibility theorem for such metrics, and as time allows describe some of the ideas involved in the proofs.<br />
<br />
===Ethan Farber===<br />
<br />
"Constructing pseudo-Anosovs from expanding interval maps"<br />
<br />
The celebrated Nielsen-Thurston classification of surface homeomorphisms says that, up to isotopy, there are three types of homeomorphisms of a closed, connected surface: (1) finite order, (2) reducible, and (3) pseudo-Anosov. Of these three types, pseudo-Anosovs are the most intriguing to dynamicists, with connections to symbolic dynamics and flat geometry. In this talk we investigate a construction of generalized pseudo-Anosovs from interval maps, first introduced by de Carvalho. In particular, for a certain class of interval maps we give necessary and sufficient conditions for the construction to produce a true pseudo-Anosov, which may be recast in terms of the kneading data of the interval map. We also describe a bijection between such interval maps and the rationals in the open unit interval which captures the kneading data, and which increases monotonically in the entropy of the interval map.<br />
<br />
===Jon Chaika===<br />
<br />
"A strange limit of horocycle ergodic measures in a stratum of<br />
translation surfaces"<br />
<br />
The main result of this talk is that in the space of unit area<br />
translation surfaces with one cone point there is a weak-star limit of<br />
measures on periodic horocycles that is fully supported in the<br />
7-dimensional space but gives positive measure to a 3-dimensional<br />
submanifold. As a consequence we obtain a non-genericity result for the<br />
horocycle flow in this space. I will define the terminology. This is joint<br />
work with Osama Khalil and John Smillie.<br />
<br />
===Harrison Bray===<br />
<br />
"Volume-entropy rigidity for convex real projective manifolds"<br />
<br />
I will discuss joint work with Constantine, building on joint work with Adeboye and Constantine, on a volume-entropy rigidity result for finite volume strictly convex projective manifolds in dimension at least 3. The result is a Besson-Courtois-Gallot type theorem, using the barycenter method. As an application, we get a uniform lower bound on the Hilbert volume of a finite volume strictly convex projective manifold of dimension at least 3.<br />
<br />
===Claire Burrin===<br />
<br />
"A sparse equidistribution problem for expanding horocycles on the modular surface"<br />
<br />
Abstract: The orbits of the horocycle flow on hyperbolic surfaces (or orbifolds) are classified: each orbit is either dense or a closed horocycle around a cusp. Expanding closed horocycles are themselves asymptotically dense, and in fact become equidistributed on the surface. The precise rate of equidistribution is of interest; on the modular surface, Zagier observed that a particular rate is equivalent to the Riemann hypothesis being true. In this talk, I will discuss the asymptotic behavior of evenly spaced points along an expanding closed horocycle on the modular surface. In this problem, the number of points depends on the expansion rate of the horocycle, and the difficulty is that these points are no more invariant under the horocycle flow. This is based on joint work with Uri Shapira and Shucheng Yu.<br />
<br />
===Kasra Rafi===<br />
<br />
"Absolutely continuous stationary measures for the mapping class group"<br />
<br />
We prove a version of a Theorem of Furstenberg in the setting of Mapping class groups. Thurston measure defines a smooth measure class on space of projectivized measured laminations For every measure \nu in this measure class, we produce a measure \mu with finite first moment on the mapping class group such that \nu is the unique \mu-stationary measure. In particular, this gives an coding-free proof of the already known result that the Lyapunov spectrum of Kontsevich-Zorich cocycle on the principal stratum of quadratic differentials is simple. This is a joint work with Alex Eskin and Maryam Mirzakhani. <br />
<br />
<br />
<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=21117Dynamics Seminar 2020-20212021-04-06T17:53:25Z<p>Cwu367: </p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
Meetings are on Zoom. To get Zoom info email Chenxi Wu. <br />
<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
== Spring 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|February 3<br />
|Daniel Woodhouse (Oxford)<br />
|Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups<br />
| <br />
|-<br />
|February 10<br />
|John Mackay (Bristol)<br />
|Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy<br />
| <br />
|-<br />
|February 17<br />
|Benjamin Branman (Wisconsin)<br />
|Spaces of Pants Decompositions for Surfaces of Infinite Type<br />
| <br />
|-<br />
|February 24<br />
|Uri Bader (Weizmann Institute)<br />
|Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity.<br />
| <br />
|-<br />
|March 3<br />
|Omri Sarig (Weizmann Institute)<br />
|(Dis)continuity of Lyapunov exponents for surface diffeomorphisms (joint with J. Buzz and S. Crovisier)<br />
| <br />
|-<br />
|March 10<br />
|Chris Leininger (Rice University)<br />
|Billiards, symbolic coding, and cone metrics<br />
| <br />
|-<br />
|March 17<br />
|Ethan Farber (Boston College)<br />
|Constructing pseudo-Anosovs from expanding interval maps<br />
|<br />
|-<br />
|March 24<br />
|Jon Chaika (Utah)<br />
|A strange limit of horocycle ergodic measures in a stratum of<br />
translation surfaces<br />
|<br />
|-<br />
|March 31<br />
|Harrison Bray (George Mason)<br />
|Volume-entropy rigidity for convex real projective manifolds<br />
|<br />
|-<br />
|April 7<br />
|Claire Burrin （ETH Zurich)<br />
|A sparse equidistribution problem for expanding horocycles on the modular surface<br />
|<br />
|-<br />
|April 21<br />
|Kasra Rafi (Toronto)<br />
|TBA<br />
|<br />
|-<br />
|April 28<br />
|Matt Bainbridge (Indiana)<br />
|TBA<br />
|}<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
===Daniel Woodhouse===<br />
<br />
"Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups"<br />
<br />
Let F be a finitely rank free group.<br />
Let w_1 and w_2 be suitable random/generic elements in F.<br />
Consider the HNN extension G = <F, t | t w_1 t^{-1} = w_2 >.<br />
It is known from existing results that G will be 1-ended and hyperbolic.<br />
We have shown that G is quasi-isometrically rigid.<br />
That is to say that if a f.g. group H is quasi-isometric to G, then G and H are virtually isomorphic.<br />
The full result is for finite graphs of groups with virtually free vertex groups and and two-ended edge groups, but the statement is more technical -- not all such groups are QI-rigid.<br />
The main argument involves applying a new proof of Leighton's graph covering theorem.<br />
This is joint work with Sam Shepherd.<br />
<br />
===John Mackay===<br />
<br />
"Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy"<br />
<br />
The separation profile of an infinite graph was introduced by<br />
Benjamini-Schramm-Timar. It is a function which measures how<br />
well-connected the graph is by how hard it is to cut finite subgraphs<br />
into small pieces. In earlier joint work with David Hume and Romain<br />
Tessera, we introduced Poincaré profiles, generalising this concept by<br />
using p-Poincaré inequalities to measure the connected-ness of<br />
subgraphs. I will discuss this family of invariants, their applications<br />
to coarse embedding problems, and recent work finding the profiles of<br />
all connected unimodular Lie groups, where a dichotomy is exhibited.<br />
Joint with Hume and Tessera.<br />
<br />
===Benjamin Branman===<br />
<br />
"Spaces of Pants Decompositions for Surfaces of Infinite Type"<br />
<br />
We study the pants graph of surfaces of infinite type. When S is a surface of infinite type, the usual definition of the graph of pants decompositions yields a graph with infinitely many connected-components. In the first part of our talk, we study this disconnected graph. In particular, we show that the extended mapping class group of S is isomorphic to a proper subgroup of of the pants graph, in contrast to the finite-type case. In the second part of the talk, motivated by the Metaconjecture of Ivanov, we seek to endow the pants graph with additional structure. To this end, we define a coarser topology on the pants graph than the topology inherited from the graph structure. We show that our new space is path-connected, and that its automorphism group is isomorphic to the extended mapping class group.<br />
<br />
<br />
===Uri Bader===<br />
"Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity."<br />
<br />
Compact hyperbolic manifolds are very interesting geometric objects.<br />
Maybe surprisingly, they are also interesting from an algebraic point of view:<br />
They are completely determined by their fundamental groups (this is Mostow's Theorem),<br />
which could be seen as a subgroup of the integer valued invertible matrices in some dimension, GL_n(Z).<br />
When the fundamental group is the Z-points of some algebraic subgroup of GL_n we say that the manifold is arithmetic.<br />
A question arises: is there a simple geometric criterion for arithmeticity for hyperbolic manifolds?<br />
Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen.<br />
In a recent work with Fisher, Miller and Stover we proved this conjecture.<br />
Our proof is based on the theory of AREA, namely Algebraic Representation of Ergodic Actions, which I have developed with Alex Furman in recent years.<br />
In this talk I will try to survey the subject in a colloquial manner.<br />
<br />
<br />
===Omri Sarig===<br />
<br />
"(Dis)continuity of Lyapunov exponents for surface diffeomorphisms" (joint with J. Buzz and S. Crovisier)"<br />
<br />
Let f be an infinitely differentiable surface diffeomorphism. Suppose we are given a sequence of ergodic invariant measures m_n which converge weak star to an ergodic limit m. What do we need to know on m_n to guarantee that the Lyapunov exponents of m_n converge to the Lyapunov exponents of m?<br />
The main result is that if m has positive entropy, and the entropy of m_n converges to the entropy of m, then the Lyapunov exponents of m_n converge to the Lyapunov exponents of m.<br />
This is joint work with J. Buzzi and S. Crovisier.<br />
<br />
===Chris Leininger===<br />
<br />
"Billiards, symbolic coding, and cone metrics"<br />
<br />
Given a polygon in the Euclidean or hyperbolic plane a billiard trajectory in the polygon is the geodesic path of a particle in the polygon bouncing off the sides so that the angle of reflection is equal to the angle incidence. A billiard trajectory determines a symbolic coding via the sides of the polygon encountered. In this talk I will describe joint work with Erlandsson and Sadanand showing the extent to which the set of all coding sequences, the bounce spectrum, determines the shape of a hyperbolic polygon. We completely characterize those polygons which are billiard rigid (the generic case), meaning that they are determined up to isometry by their bounce spectrum. When rigidity fails for a polygon P, we parameterize the space of polygons having the same bounce spectrum at P. These results for billiards are a consequence of a rigidity/flexibility theorem for negatively curved hyperbolic cone metrics. In the talk I will explain the theorem about hyperbolic billiards, comparing/contrasting it with the Euclidean case (earlier work with Duchin, Erlandsson, and Sadanand). Then I will explain the relationship with hyperbolic cone metrics, state our rigidity/flexibility theorem for such metrics, and as time allows describe some of the ideas involved in the proofs.<br />
<br />
===Ethan Farber===<br />
<br />
"Constructing pseudo-Anosovs from expanding interval maps"<br />
<br />
The celebrated Nielsen-Thurston classification of surface homeomorphisms says that, up to isotopy, there are three types of homeomorphisms of a closed, connected surface: (1) finite order, (2) reducible, and (3) pseudo-Anosov. Of these three types, pseudo-Anosovs are the most intriguing to dynamicists, with connections to symbolic dynamics and flat geometry. In this talk we investigate a construction of generalized pseudo-Anosovs from interval maps, first introduced by de Carvalho. In particular, for a certain class of interval maps we give necessary and sufficient conditions for the construction to produce a true pseudo-Anosov, which may be recast in terms of the kneading data of the interval map. We also describe a bijection between such interval maps and the rationals in the open unit interval which captures the kneading data, and which increases monotonically in the entropy of the interval map.<br />
<br />
===Jon Chaika===<br />
<br />
"A strange limit of horocycle ergodic measures in a stratum of<br />
translation surfaces"<br />
<br />
The main result of this talk is that in the space of unit area<br />
translation surfaces with one cone point there is a weak-star limit of<br />
measures on periodic horocycles that is fully supported in the<br />
7-dimensional space but gives positive measure to a 3-dimensional<br />
submanifold. As a consequence we obtain a non-genericity result for the<br />
horocycle flow in this space. I will define the terminology. This is joint<br />
work with Osama Khalil and John Smillie.<br />
<br />
===Harrison Bray===<br />
<br />
"Volume-entropy rigidity for convex real projective manifolds"<br />
<br />
I will discuss joint work with Constantine, building on joint work with Adeboye and Constantine, on a volume-entropy rigidity result for finite volume strictly convex projective manifolds in dimension at least 3. The result is a Besson-Courtois-Gallot type theorem, using the barycenter method. As an application, we get a uniform lower bound on the Hilbert volume of a finite volume strictly convex projective manifold of dimension at least 3.<br />
<br />
===Claire Burrin===<br />
<br />
"A sparse equidistribution problem for expanding horocycles on the modular surface"<br />
<br />
Abstract: The orbits of the horocycle flow on hyperbolic surfaces (or orbifolds) are classified: each orbit is either dense or a closed horocycle around a cusp. Expanding closed horocycles are themselves asymptotically dense, and in fact become equidistributed on the surface. The precise rate of equidistribution is of interest; on the modular surface, Zagier observed that a particular rate is equivalent to the Riemann hypothesis being true. In this talk, I will discuss the asymptotic behavior of evenly spaced points along an expanding closed horocycle on the modular surface. In this problem, the number of points depends on the expansion rate of the horocycle, and the difficulty is that these points are no more invariant under the horocycle flow. This is based on joint work with Uri Shapira and Shucheng Yu.<br />
<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=21084Dynamics Seminar 2020-20212021-03-30T15:46:03Z<p>Cwu367: </p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
Meetings are on Zoom. To get Zoom info email Chenxi Wu. <br />
<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
== Spring 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|February 3<br />
|Daniel Woodhouse (Oxford)<br />
|Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups<br />
| <br />
|-<br />
|February 10<br />
|John Mackay (Bristol)<br />
|Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy<br />
| <br />
|-<br />
|February 17<br />
|Benjamin Branman (Wisconsin)<br />
|Spaces of Pants Decompositions for Surfaces of Infinite Type<br />
| <br />
|-<br />
|February 24<br />
|Uri Bader (Weizmann Institute)<br />
|Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity.<br />
| <br />
|-<br />
|March 3<br />
|Omri Sarig (Weizmann Institute)<br />
|(Dis)continuity of Lyapunov exponents for surface diffeomorphisms (joint with J. Buzz and S. Crovisier)<br />
| <br />
|-<br />
|March 10<br />
|Chris Leininger (Rice University)<br />
|Billiards, symbolic coding, and cone metrics<br />
| <br />
|-<br />
|March 17<br />
|Ethan Farber (Boston College)<br />
|Constructing pseudo-Anosovs from expanding interval maps<br />
|<br />
|-<br />
|March 24<br />
|Jon Chaika (Utah)<br />
|A strange limit of horocycle ergodic measures in a stratum of<br />
translation surfaces<br />
|<br />
|-<br />
|March 31<br />
|Harrison Bray (George Mason)<br />
|Volume-entropy rigidity for convex real projective manifolds<br />
|<br />
|-<br />
|April 7<br />
|Claire Burrin （ETH Zurich)<br />
|TBA<br />
|<br />
|-<br />
|April 21<br />
|Kasra Rafi (Toronto)<br />
|TBA<br />
|<br />
|-<br />
|April 28<br />
|Matt Bainbridge (Indiana)<br />
|TBA<br />
|}<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
===Daniel Woodhouse===<br />
<br />
"Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups"<br />
<br />
Let F be a finitely rank free group.<br />
Let w_1 and w_2 be suitable random/generic elements in F.<br />
Consider the HNN extension G = <F, t | t w_1 t^{-1} = w_2 >.<br />
It is known from existing results that G will be 1-ended and hyperbolic.<br />
We have shown that G is quasi-isometrically rigid.<br />
That is to say that if a f.g. group H is quasi-isometric to G, then G and H are virtually isomorphic.<br />
The full result is for finite graphs of groups with virtually free vertex groups and and two-ended edge groups, but the statement is more technical -- not all such groups are QI-rigid.<br />
The main argument involves applying a new proof of Leighton's graph covering theorem.<br />
This is joint work with Sam Shepherd.<br />
<br />
===John Mackay===<br />
<br />
"Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy"<br />
<br />
The separation profile of an infinite graph was introduced by<br />
Benjamini-Schramm-Timar. It is a function which measures how<br />
well-connected the graph is by how hard it is to cut finite subgraphs<br />
into small pieces. In earlier joint work with David Hume and Romain<br />
Tessera, we introduced Poincaré profiles, generalising this concept by<br />
using p-Poincaré inequalities to measure the connected-ness of<br />
subgraphs. I will discuss this family of invariants, their applications<br />
to coarse embedding problems, and recent work finding the profiles of<br />
all connected unimodular Lie groups, where a dichotomy is exhibited.<br />
Joint with Hume and Tessera.<br />
<br />
===Benjamin Branman===<br />
<br />
"Spaces of Pants Decompositions for Surfaces of Infinite Type"<br />
<br />
We study the pants graph of surfaces of infinite type. When S is a surface of infinite type, the usual definition of the graph of pants decompositions yields a graph with infinitely many connected-components. In the first part of our talk, we study this disconnected graph. In particular, we show that the extended mapping class group of S is isomorphic to a proper subgroup of of the pants graph, in contrast to the finite-type case. In the second part of the talk, motivated by the Metaconjecture of Ivanov, we seek to endow the pants graph with additional structure. To this end, we define a coarser topology on the pants graph than the topology inherited from the graph structure. We show that our new space is path-connected, and that its automorphism group is isomorphic to the extended mapping class group.<br />
<br />
<br />
===Uri Bader===<br />
"Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity."<br />
<br />
Compact hyperbolic manifolds are very interesting geometric objects.<br />
Maybe surprisingly, they are also interesting from an algebraic point of view:<br />
They are completely determined by their fundamental groups (this is Mostow's Theorem),<br />
which could be seen as a subgroup of the integer valued invertible matrices in some dimension, GL_n(Z).<br />
When the fundamental group is the Z-points of some algebraic subgroup of GL_n we say that the manifold is arithmetic.<br />
A question arises: is there a simple geometric criterion for arithmeticity for hyperbolic manifolds?<br />
Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen.<br />
In a recent work with Fisher, Miller and Stover we proved this conjecture.<br />
Our proof is based on the theory of AREA, namely Algebraic Representation of Ergodic Actions, which I have developed with Alex Furman in recent years.<br />
In this talk I will try to survey the subject in a colloquial manner.<br />
<br />
<br />
===Omri Sarig===<br />
<br />
"(Dis)continuity of Lyapunov exponents for surface diffeomorphisms" (joint with J. Buzz and S. Crovisier)"<br />
<br />
Let f be an infinitely differentiable surface diffeomorphism. Suppose we are given a sequence of ergodic invariant measures m_n which converge weak star to an ergodic limit m. What do we need to know on m_n to guarantee that the Lyapunov exponents of m_n converge to the Lyapunov exponents of m?<br />
The main result is that if m has positive entropy, and the entropy of m_n converges to the entropy of m, then the Lyapunov exponents of m_n converge to the Lyapunov exponents of m.<br />
This is joint work with J. Buzzi and S. Crovisier.<br />
<br />
===Chris Leininger===<br />
<br />
"Billiards, symbolic coding, and cone metrics"<br />
<br />
Given a polygon in the Euclidean or hyperbolic plane a billiard trajectory in the polygon is the geodesic path of a particle in the polygon bouncing off the sides so that the angle of reflection is equal to the angle incidence. A billiard trajectory determines a symbolic coding via the sides of the polygon encountered. In this talk I will describe joint work with Erlandsson and Sadanand showing the extent to which the set of all coding sequences, the bounce spectrum, determines the shape of a hyperbolic polygon. We completely characterize those polygons which are billiard rigid (the generic case), meaning that they are determined up to isometry by their bounce spectrum. When rigidity fails for a polygon P, we parameterize the space of polygons having the same bounce spectrum at P. These results for billiards are a consequence of a rigidity/flexibility theorem for negatively curved hyperbolic cone metrics. In the talk I will explain the theorem about hyperbolic billiards, comparing/contrasting it with the Euclidean case (earlier work with Duchin, Erlandsson, and Sadanand). Then I will explain the relationship with hyperbolic cone metrics, state our rigidity/flexibility theorem for such metrics, and as time allows describe some of the ideas involved in the proofs.<br />
<br />
===Ethan Farber===<br />
<br />
"Constructing pseudo-Anosovs from expanding interval maps"<br />
<br />
The celebrated Nielsen-Thurston classification of surface homeomorphisms says that, up to isotopy, there are three types of homeomorphisms of a closed, connected surface: (1) finite order, (2) reducible, and (3) pseudo-Anosov. Of these three types, pseudo-Anosovs are the most intriguing to dynamicists, with connections to symbolic dynamics and flat geometry. In this talk we investigate a construction of generalized pseudo-Anosovs from interval maps, first introduced by de Carvalho. In particular, for a certain class of interval maps we give necessary and sufficient conditions for the construction to produce a true pseudo-Anosov, which may be recast in terms of the kneading data of the interval map. We also describe a bijection between such interval maps and the rationals in the open unit interval which captures the kneading data, and which increases monotonically in the entropy of the interval map.<br />
<br />
===Jon Chaika===<br />
<br />
"A strange limit of horocycle ergodic measures in a stratum of<br />
translation surfaces"<br />
<br />
The main result of this talk is that in the space of unit area<br />
translation surfaces with one cone point there is a weak-star limit of<br />
measures on periodic horocycles that is fully supported in the<br />
7-dimensional space but gives positive measure to a 3-dimensional<br />
submanifold. As a consequence we obtain a non-genericity result for the<br />
horocycle flow in this space. I will define the terminology. This is joint<br />
work with Osama Khalil and John Smillie.<br />
<br />
===Harrison Bray===<br />
<br />
"Volume-entropy rigidity for convex real projective manifolds"<br />
<br />
I will discuss joint work with Constantine, building on joint work with Adeboye and Constantine, on a volume-entropy rigidity result for finite volume strictly convex projective manifolds in dimension at least 3. The result is a Besson-Courtois-Gallot type theorem, using the barycenter method. As an application, we get a uniform lower bound on the Hilbert volume of a finite volume strictly convex projective manifold of dimension at least 3.<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=21051Dynamics Seminar 2020-20212021-03-23T21:27:29Z<p>Cwu367: /* Spring 2021 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
Meetings are on Zoom. To get Zoom info email Chenxi Wu. <br />
<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
== Spring 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|February 3<br />
|Daniel Woodhouse (Oxford)<br />
|Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups<br />
| <br />
|-<br />
|February 10<br />
|John Mackay (Bristol)<br />
|Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy<br />
| <br />
|-<br />
|February 17<br />
|Benjamin Branman (Wisconsin)<br />
|Spaces of Pants Decompositions for Surfaces of Infinite Type<br />
| <br />
|-<br />
|February 24<br />
|Uri Bader (Weizmann Institute)<br />
|Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity.<br />
| <br />
|-<br />
|March 3<br />
|Omri Sarig (Weizmann Institute)<br />
|(Dis)continuity of Lyapunov exponents for surface diffeomorphisms (joint with J. Buzz and S. Crovisier)<br />
| <br />
|-<br />
|March 10<br />
|Chris Leininger (Rice University)<br />
|Billiards, symbolic coding, and cone metrics<br />
| <br />
|-<br />
|March 17<br />
|Ethan Farber (Boston College)<br />
|Constructing pseudo-Anosovs from expanding interval maps<br />
|<br />
|-<br />
|March 24<br />
|Jon Chaika (Utah)<br />
|A strange limit of horocycle ergodic measures in a stratum of<br />
translation surfaces<br />
|<br />
|-<br />
|March 31<br />
|Harrison Bray (George Mason)<br />
|TBA<br />
|<br />
|-<br />
|April 7<br />
|Claire Burrin （ETH Zurich)<br />
|TBA<br />
|<br />
|-<br />
|April 21<br />
|Kasra Rafi (Toronto)<br />
|TBA<br />
|<br />
|-<br />
|April 28<br />
|Matt Bainbridge (Indiana)<br />
|TBA<br />
|}<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
===Daniel Woodhouse===<br />
<br />
"Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups"<br />
<br />
Let F be a finitely rank free group.<br />
Let w_1 and w_2 be suitable random/generic elements in F.<br />
Consider the HNN extension G = <F, t | t w_1 t^{-1} = w_2 >.<br />
It is known from existing results that G will be 1-ended and hyperbolic.<br />
We have shown that G is quasi-isometrically rigid.<br />
That is to say that if a f.g. group H is quasi-isometric to G, then G and H are virtually isomorphic.<br />
The full result is for finite graphs of groups with virtually free vertex groups and and two-ended edge groups, but the statement is more technical -- not all such groups are QI-rigid.<br />
The main argument involves applying a new proof of Leighton's graph covering theorem.<br />
This is joint work with Sam Shepherd.<br />
<br />
===John Mackay===<br />
<br />
"Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy"<br />
<br />
The separation profile of an infinite graph was introduced by<br />
Benjamini-Schramm-Timar. It is a function which measures how<br />
well-connected the graph is by how hard it is to cut finite subgraphs<br />
into small pieces. In earlier joint work with David Hume and Romain<br />
Tessera, we introduced Poincaré profiles, generalising this concept by<br />
using p-Poincaré inequalities to measure the connected-ness of<br />
subgraphs. I will discuss this family of invariants, their applications<br />
to coarse embedding problems, and recent work finding the profiles of<br />
all connected unimodular Lie groups, where a dichotomy is exhibited.<br />
Joint with Hume and Tessera.<br />
<br />
===Benjamin Branman===<br />
<br />
"Spaces of Pants Decompositions for Surfaces of Infinite Type"<br />
<br />
We study the pants graph of surfaces of infinite type. When S is a surface of infinite type, the usual definition of the graph of pants decompositions yields a graph with infinitely many connected-components. In the first part of our talk, we study this disconnected graph. In particular, we show that the extended mapping class group of S is isomorphic to a proper subgroup of of the pants graph, in contrast to the finite-type case. In the second part of the talk, motivated by the Metaconjecture of Ivanov, we seek to endow the pants graph with additional structure. To this end, we define a coarser topology on the pants graph than the topology inherited from the graph structure. We show that our new space is path-connected, and that its automorphism group is isomorphic to the extended mapping class group.<br />
<br />
<br />
===Uri Bader===<br />
"Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity."<br />
<br />
Compact hyperbolic manifolds are very interesting geometric objects.<br />
Maybe surprisingly, they are also interesting from an algebraic point of view:<br />
They are completely determined by their fundamental groups (this is Mostow's Theorem),<br />
which could be seen as a subgroup of the integer valued invertible matrices in some dimension, GL_n(Z).<br />
When the fundamental group is the Z-points of some algebraic subgroup of GL_n we say that the manifold is arithmetic.<br />
A question arises: is there a simple geometric criterion for arithmeticity for hyperbolic manifolds?<br />
Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen.<br />
In a recent work with Fisher, Miller and Stover we proved this conjecture.<br />
Our proof is based on the theory of AREA, namely Algebraic Representation of Ergodic Actions, which I have developed with Alex Furman in recent years.<br />
In this talk I will try to survey the subject in a colloquial manner.<br />
<br />
<br />
===Omri Sarig===<br />
<br />
"(Dis)continuity of Lyapunov exponents for surface diffeomorphisms" (joint with J. Buzz and S. Crovisier)"<br />
<br />
Let f be an infinitely differentiable surface diffeomorphism. Suppose we are given a sequence of ergodic invariant measures m_n which converge weak star to an ergodic limit m. What do we need to know on m_n to guarantee that the Lyapunov exponents of m_n converge to the Lyapunov exponents of m?<br />
The main result is that if m has positive entropy, and the entropy of m_n converges to the entropy of m, then the Lyapunov exponents of m_n converge to the Lyapunov exponents of m.<br />
This is joint work with J. Buzzi and S. Crovisier.<br />
<br />
===Chris Leininger===<br />
<br />
"Billiards, symbolic coding, and cone metrics"<br />
<br />
Given a polygon in the Euclidean or hyperbolic plane a billiard trajectory in the polygon is the geodesic path of a particle in the polygon bouncing off the sides so that the angle of reflection is equal to the angle incidence. A billiard trajectory determines a symbolic coding via the sides of the polygon encountered. In this talk I will describe joint work with Erlandsson and Sadanand showing the extent to which the set of all coding sequences, the bounce spectrum, determines the shape of a hyperbolic polygon. We completely characterize those polygons which are billiard rigid (the generic case), meaning that they are determined up to isometry by their bounce spectrum. When rigidity fails for a polygon P, we parameterize the space of polygons having the same bounce spectrum at P. These results for billiards are a consequence of a rigidity/flexibility theorem for negatively curved hyperbolic cone metrics. In the talk I will explain the theorem about hyperbolic billiards, comparing/contrasting it with the Euclidean case (earlier work with Duchin, Erlandsson, and Sadanand). Then I will explain the relationship with hyperbolic cone metrics, state our rigidity/flexibility theorem for such metrics, and as time allows describe some of the ideas involved in the proofs.<br />
<br />
===Ethan Farber===<br />
<br />
"Constructing pseudo-Anosovs from expanding interval maps"<br />
<br />
The celebrated Nielsen-Thurston classification of surface homeomorphisms says that, up to isotopy, there are three types of homeomorphisms of a closed, connected surface: (1) finite order, (2) reducible, and (3) pseudo-Anosov. Of these three types, pseudo-Anosovs are the most intriguing to dynamicists, with connections to symbolic dynamics and flat geometry. In this talk we investigate a construction of generalized pseudo-Anosovs from interval maps, first introduced by de Carvalho. In particular, for a certain class of interval maps we give necessary and sufficient conditions for the construction to produce a true pseudo-Anosov, which may be recast in terms of the kneading data of the interval map. We also describe a bijection between such interval maps and the rationals in the open unit interval which captures the kneading data, and which increases monotonically in the entropy of the interval map.<br />
<br />
===Jon Chaika===<br />
<br />
"A strange limit of horocycle ergodic measures in a stratum of<br />
translation surfaces"<br />
<br />
The main result of this talk is that in the space of unit area<br />
translation surfaces with one cone point there is a weak-star limit of<br />
measures on periodic horocycles that is fully supported in the<br />
7-dimensional space but gives positive measure to a 3-dimensional<br />
submanifold. As a consequence we obtain a non-genericity result for the<br />
horocycle flow in this space. I will define the terminology. This is joint<br />
work with Osama Khalil and John Smillie.<br />
<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=21050Dynamics Seminar 2020-20212021-03-23T21:26:52Z<p>Cwu367: </p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
Meetings are on Zoom. To get Zoom info email Chenxi Wu. <br />
<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
== Spring 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|February 3<br />
|Daniel Woodhouse (Oxford)<br />
|Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups<br />
| <br />
|-<br />
|February 10<br />
|John Mackay (Bristol)<br />
|Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy<br />
| <br />
|-<br />
|February 17<br />
|Benjamin Branman (Wisconsin)<br />
|Spaces of Pants Decompositions for Surfaces of Infinite Type<br />
| <br />
|-<br />
|February 24<br />
|Uri Bader (Weizmann Institute)<br />
|Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity.<br />
| <br />
|-<br />
|March 3<br />
|Omri Sarig (Weizmann Institute)<br />
|(Dis)continuity of Lyapunov exponents for surface diffeomorphisms (joint with J. Buzz and S. Crovisier)<br />
| <br />
|-<br />
|March 10<br />
|Chris Leininger (Rice University)<br />
|Billiards, symbolic coding, and cone metrics<br />
| <br />
|-<br />
|March 17<br />
|Ethan Farber (Boston College)<br />
|Constructing pseudo-Anosovs from expanding interval maps<br />
|<br />
|-<br />
|March 24<br />
|Jon Chaika (Utah)<br />
|strange limit of horocycle ergodic measures in a stratum of<br />
translation surfaces<br />
|<br />
|-<br />
|March 31<br />
|Harrison Bray (George Mason)<br />
|TBA<br />
|<br />
|-<br />
|April 7<br />
|Claire Burrin （ETH Zurich)<br />
|TBA<br />
|<br />
|-<br />
|April 21<br />
|Kasra Rafi (Toronto)<br />
|TBA<br />
|<br />
|-<br />
|April 28<br />
|Matt Bainbridge (Indiana)<br />
|TBA<br />
|}<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
===Daniel Woodhouse===<br />
<br />
"Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups"<br />
<br />
Let F be a finitely rank free group.<br />
Let w_1 and w_2 be suitable random/generic elements in F.<br />
Consider the HNN extension G = <F, t | t w_1 t^{-1} = w_2 >.<br />
It is known from existing results that G will be 1-ended and hyperbolic.<br />
We have shown that G is quasi-isometrically rigid.<br />
That is to say that if a f.g. group H is quasi-isometric to G, then G and H are virtually isomorphic.<br />
The full result is for finite graphs of groups with virtually free vertex groups and and two-ended edge groups, but the statement is more technical -- not all such groups are QI-rigid.<br />
The main argument involves applying a new proof of Leighton's graph covering theorem.<br />
This is joint work with Sam Shepherd.<br />
<br />
===John Mackay===<br />
<br />
"Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy"<br />
<br />
The separation profile of an infinite graph was introduced by<br />
Benjamini-Schramm-Timar. It is a function which measures how<br />
well-connected the graph is by how hard it is to cut finite subgraphs<br />
into small pieces. In earlier joint work with David Hume and Romain<br />
Tessera, we introduced Poincaré profiles, generalising this concept by<br />
using p-Poincaré inequalities to measure the connected-ness of<br />
subgraphs. I will discuss this family of invariants, their applications<br />
to coarse embedding problems, and recent work finding the profiles of<br />
all connected unimodular Lie groups, where a dichotomy is exhibited.<br />
Joint with Hume and Tessera.<br />
<br />
===Benjamin Branman===<br />
<br />
"Spaces of Pants Decompositions for Surfaces of Infinite Type"<br />
<br />
We study the pants graph of surfaces of infinite type. When S is a surface of infinite type, the usual definition of the graph of pants decompositions yields a graph with infinitely many connected-components. In the first part of our talk, we study this disconnected graph. In particular, we show that the extended mapping class group of S is isomorphic to a proper subgroup of of the pants graph, in contrast to the finite-type case. In the second part of the talk, motivated by the Metaconjecture of Ivanov, we seek to endow the pants graph with additional structure. To this end, we define a coarser topology on the pants graph than the topology inherited from the graph structure. We show that our new space is path-connected, and that its automorphism group is isomorphic to the extended mapping class group.<br />
<br />
<br />
===Uri Bader===<br />
"Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity."<br />
<br />
Compact hyperbolic manifolds are very interesting geometric objects.<br />
Maybe surprisingly, they are also interesting from an algebraic point of view:<br />
They are completely determined by their fundamental groups (this is Mostow's Theorem),<br />
which could be seen as a subgroup of the integer valued invertible matrices in some dimension, GL_n(Z).<br />
When the fundamental group is the Z-points of some algebraic subgroup of GL_n we say that the manifold is arithmetic.<br />
A question arises: is there a simple geometric criterion for arithmeticity for hyperbolic manifolds?<br />
Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen.<br />
In a recent work with Fisher, Miller and Stover we proved this conjecture.<br />
Our proof is based on the theory of AREA, namely Algebraic Representation of Ergodic Actions, which I have developed with Alex Furman in recent years.<br />
In this talk I will try to survey the subject in a colloquial manner.<br />
<br />
<br />
===Omri Sarig===<br />
<br />
"(Dis)continuity of Lyapunov exponents for surface diffeomorphisms" (joint with J. Buzz and S. Crovisier)"<br />
<br />
Let f be an infinitely differentiable surface diffeomorphism. Suppose we are given a sequence of ergodic invariant measures m_n which converge weak star to an ergodic limit m. What do we need to know on m_n to guarantee that the Lyapunov exponents of m_n converge to the Lyapunov exponents of m?<br />
The main result is that if m has positive entropy, and the entropy of m_n converges to the entropy of m, then the Lyapunov exponents of m_n converge to the Lyapunov exponents of m.<br />
This is joint work with J. Buzzi and S. Crovisier.<br />
<br />
===Chris Leininger===<br />
<br />
"Billiards, symbolic coding, and cone metrics"<br />
<br />
Given a polygon in the Euclidean or hyperbolic plane a billiard trajectory in the polygon is the geodesic path of a particle in the polygon bouncing off the sides so that the angle of reflection is equal to the angle incidence. A billiard trajectory determines a symbolic coding via the sides of the polygon encountered. In this talk I will describe joint work with Erlandsson and Sadanand showing the extent to which the set of all coding sequences, the bounce spectrum, determines the shape of a hyperbolic polygon. We completely characterize those polygons which are billiard rigid (the generic case), meaning that they are determined up to isometry by their bounce spectrum. When rigidity fails for a polygon P, we parameterize the space of polygons having the same bounce spectrum at P. These results for billiards are a consequence of a rigidity/flexibility theorem for negatively curved hyperbolic cone metrics. In the talk I will explain the theorem about hyperbolic billiards, comparing/contrasting it with the Euclidean case (earlier work with Duchin, Erlandsson, and Sadanand). Then I will explain the relationship with hyperbolic cone metrics, state our rigidity/flexibility theorem for such metrics, and as time allows describe some of the ideas involved in the proofs.<br />
<br />
===Ethan Farber===<br />
<br />
"Constructing pseudo-Anosovs from expanding interval maps"<br />
<br />
The celebrated Nielsen-Thurston classification of surface homeomorphisms says that, up to isotopy, there are three types of homeomorphisms of a closed, connected surface: (1) finite order, (2) reducible, and (3) pseudo-Anosov. Of these three types, pseudo-Anosovs are the most intriguing to dynamicists, with connections to symbolic dynamics and flat geometry. In this talk we investigate a construction of generalized pseudo-Anosovs from interval maps, first introduced by de Carvalho. In particular, for a certain class of interval maps we give necessary and sufficient conditions for the construction to produce a true pseudo-Anosov, which may be recast in terms of the kneading data of the interval map. We also describe a bijection between such interval maps and the rationals in the open unit interval which captures the kneading data, and which increases monotonically in the entropy of the interval map.<br />
<br />
===Jon Chaika===<br />
<br />
"A strange limit of horocycle ergodic measures in a stratum of<br />
translation surfaces"<br />
<br />
The main result of this talk is that in the space of unit area<br />
translation surfaces with one cone point there is a weak-star limit of<br />
measures on periodic horocycles that is fully supported in the<br />
7-dimensional space but gives positive measure to a 3-dimensional<br />
submanifold. As a consequence we obtain a non-genericity result for the<br />
horocycle flow in this space. I will define the terminology. This is joint<br />
work with Osama Khalil and John Smillie.<br />
<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=21005Dynamics Seminar 2020-20212021-03-16T20:15:16Z<p>Cwu367: </p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
Meetings are on Zoom. To get Zoom info email Chenxi Wu. <br />
<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
== Spring 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|February 3<br />
|Daniel Woodhouse (Oxford)<br />
|Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups<br />
| <br />
|-<br />
|February 10<br />
|John Mackay (Bristol)<br />
|Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy<br />
| <br />
|-<br />
|February 17<br />
|Benjamin Branman (Wisconsin)<br />
|Spaces of Pants Decompositions for Surfaces of Infinite Type<br />
| <br />
|-<br />
|February 24<br />
|Uri Bader (Weizmann Institute)<br />
|Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity.<br />
| <br />
|-<br />
|March 3<br />
|Omri Sarig (Weizmann Institute)<br />
|(Dis)continuity of Lyapunov exponents for surface diffeomorphisms (joint with J. Buzz and S. Crovisier)<br />
| <br />
|-<br />
|March 10<br />
|Chris Leininger (Rice University)<br />
|Billiards, symbolic coding, and cone metrics<br />
| <br />
|-<br />
|March 17<br />
|Ethan Farber (Boston College)<br />
|Constructing pseudo-Anosovs from expanding interval maps<br />
|<br />
|-<br />
|March 24<br />
|Jon Chaika (Utah)<br />
|TBA<br />
|<br />
|-<br />
|March 31<br />
|Harrison Bray (George Mason)<br />
|TBA<br />
|<br />
|-<br />
|April 7<br />
|Claire Burrin （ETH Zurich)<br />
|TBA<br />
|<br />
|-<br />
|April 21<br />
|Kasra Rafi (Toronto)<br />
|TBA<br />
|<br />
|-<br />
|April 28<br />
|Matt Bainbridge (Indiana)<br />
|TBA<br />
|}<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
===Daniel Woodhouse===<br />
<br />
"Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups"<br />
<br />
Let F be a finitely rank free group.<br />
Let w_1 and w_2 be suitable random/generic elements in F.<br />
Consider the HNN extension G = <F, t | t w_1 t^{-1} = w_2 >.<br />
It is known from existing results that G will be 1-ended and hyperbolic.<br />
We have shown that G is quasi-isometrically rigid.<br />
That is to say that if a f.g. group H is quasi-isometric to G, then G and H are virtually isomorphic.<br />
The full result is for finite graphs of groups with virtually free vertex groups and and two-ended edge groups, but the statement is more technical -- not all such groups are QI-rigid.<br />
The main argument involves applying a new proof of Leighton's graph covering theorem.<br />
This is joint work with Sam Shepherd.<br />
<br />
===John Mackay===<br />
<br />
"Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy"<br />
<br />
The separation profile of an infinite graph was introduced by<br />
Benjamini-Schramm-Timar. It is a function which measures how<br />
well-connected the graph is by how hard it is to cut finite subgraphs<br />
into small pieces. In earlier joint work with David Hume and Romain<br />
Tessera, we introduced Poincaré profiles, generalising this concept by<br />
using p-Poincaré inequalities to measure the connected-ness of<br />
subgraphs. I will discuss this family of invariants, their applications<br />
to coarse embedding problems, and recent work finding the profiles of<br />
all connected unimodular Lie groups, where a dichotomy is exhibited.<br />
Joint with Hume and Tessera.<br />
<br />
===Benjamin Branman===<br />
<br />
"Spaces of Pants Decompositions for Surfaces of Infinite Type"<br />
<br />
We study the pants graph of surfaces of infinite type. When S is a surface of infinite type, the usual definition of the graph of pants decompositions yields a graph with infinitely many connected-components. In the first part of our talk, we study this disconnected graph. In particular, we show that the extended mapping class group of S is isomorphic to a proper subgroup of of the pants graph, in contrast to the finite-type case. In the second part of the talk, motivated by the Metaconjecture of Ivanov, we seek to endow the pants graph with additional structure. To this end, we define a coarser topology on the pants graph than the topology inherited from the graph structure. We show that our new space is path-connected, and that its automorphism group is isomorphic to the extended mapping class group.<br />
<br />
<br />
===Uri Bader===<br />
"Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity."<br />
<br />
Compact hyperbolic manifolds are very interesting geometric objects.<br />
Maybe surprisingly, they are also interesting from an algebraic point of view:<br />
They are completely determined by their fundamental groups (this is Mostow's Theorem),<br />
which could be seen as a subgroup of the integer valued invertible matrices in some dimension, GL_n(Z).<br />
When the fundamental group is the Z-points of some algebraic subgroup of GL_n we say that the manifold is arithmetic.<br />
A question arises: is there a simple geometric criterion for arithmeticity for hyperbolic manifolds?<br />
Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen.<br />
In a recent work with Fisher, Miller and Stover we proved this conjecture.<br />
Our proof is based on the theory of AREA, namely Algebraic Representation of Ergodic Actions, which I have developed with Alex Furman in recent years.<br />
In this talk I will try to survey the subject in a colloquial manner.<br />
<br />
<br />
===Omri Sarig===<br />
<br />
"(Dis)continuity of Lyapunov exponents for surface diffeomorphisms" (joint with J. Buzz and S. Crovisier)"<br />
<br />
Let f be an infinitely differentiable surface diffeomorphism. Suppose we are given a sequence of ergodic invariant measures m_n which converge weak star to an ergodic limit m. What do we need to know on m_n to guarantee that the Lyapunov exponents of m_n converge to the Lyapunov exponents of m?<br />
The main result is that if m has positive entropy, and the entropy of m_n converges to the entropy of m, then the Lyapunov exponents of m_n converge to the Lyapunov exponents of m.<br />
This is joint work with J. Buzzi and S. Crovisier.<br />
<br />
===Chris Leininger===<br />
<br />
"Billiards, symbolic coding, and cone metrics"<br />
<br />
Given a polygon in the Euclidean or hyperbolic plane a billiard trajectory in the polygon is the geodesic path of a particle in the polygon bouncing off the sides so that the angle of reflection is equal to the angle incidence. A billiard trajectory determines a symbolic coding via the sides of the polygon encountered. In this talk I will describe joint work with Erlandsson and Sadanand showing the extent to which the set of all coding sequences, the bounce spectrum, determines the shape of a hyperbolic polygon. We completely characterize those polygons which are billiard rigid (the generic case), meaning that they are determined up to isometry by their bounce spectrum. When rigidity fails for a polygon P, we parameterize the space of polygons having the same bounce spectrum at P. These results for billiards are a consequence of a rigidity/flexibility theorem for negatively curved hyperbolic cone metrics. In the talk I will explain the theorem about hyperbolic billiards, comparing/contrasting it with the Euclidean case (earlier work with Duchin, Erlandsson, and Sadanand). Then I will explain the relationship with hyperbolic cone metrics, state our rigidity/flexibility theorem for such metrics, and as time allows describe some of the ideas involved in the proofs.<br />
<br />
===Ethan Farber===<br />
<br />
"Constructing pseudo-Anosovs from expanding interval maps"<br />
<br />
The celebrated Nielsen-Thurston classification of surface homeomorphisms says that, up to isotopy, there are three types of homeomorphisms of a closed, connected surface: (1) finite order, (2) reducible, and (3) pseudo-Anosov. Of these three types, pseudo-Anosovs are the most intriguing to dynamicists, with connections to symbolic dynamics and flat geometry. In this talk we investigate a construction of generalized pseudo-Anosovs from interval maps, first introduced by de Carvalho. In particular, for a certain class of interval maps we give necessary and sufficient conditions for the construction to produce a true pseudo-Anosov, which may be recast in terms of the kneading data of the interval map. We also describe a bijection between such interval maps and the rationals in the open unit interval which captures the kneading data, and which increases monotonically in the entropy of the interval map.<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20965Dynamics Seminar 2020-20212021-03-09T14:16:10Z<p>Cwu367: </p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
Meetings are on Zoom. To get Zoom info email Chenxi Wu. <br />
<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
== Spring 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|February 3<br />
|Daniel Woodhouse (Oxford)<br />
|Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups<br />
| <br />
|-<br />
|February 10<br />
|John Mackay (Bristol)<br />
|Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy<br />
| <br />
|-<br />
|February 17<br />
|Benjamin Branman (Wisconsin)<br />
|Spaces of Pants Decompositions for Surfaces of Infinite Type<br />
| <br />
|-<br />
|February 24<br />
|Uri Bader (Weizmann Institute)<br />
|Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity.<br />
| <br />
|-<br />
|March 3<br />
|Omri Sarig (Weizmann Institute)<br />
|(Dis)continuity of Lyapunov exponents for surface diffeomorphisms (joint with J. Buzz and S. Crovisier)<br />
| <br />
|-<br />
|March 10<br />
|Chris Leininger (Rice University)<br />
|Billiards, symbolic coding, and cone metrics<br />
| <br />
|-<br />
|March 17<br />
|Ethan Farber (Boston College)<br />
|TBA<br />
|<br />
|-<br />
|March 24<br />
|Jon Chaika (Utah)<br />
|TBA<br />
|<br />
|-<br />
|March 31<br />
|Harrison Bray (George Mason)<br />
|TBA<br />
|<br />
|-<br />
|April 7<br />
|Claire Burrin （ETH Zurich)<br />
|TBA<br />
|<br />
|-<br />
|April 21<br />
|Kasra Rafi (Toronto)<br />
|TBA<br />
|<br />
|-<br />
|April 28<br />
|Matt Bainbridge (Indiana)<br />
|TBA<br />
|}<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
===Daniel Woodhouse===<br />
<br />
"Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups"<br />
<br />
Let F be a finitely rank free group.<br />
Let w_1 and w_2 be suitable random/generic elements in F.<br />
Consider the HNN extension G = <F, t | t w_1 t^{-1} = w_2 >.<br />
It is known from existing results that G will be 1-ended and hyperbolic.<br />
We have shown that G is quasi-isometrically rigid.<br />
That is to say that if a f.g. group H is quasi-isometric to G, then G and H are virtually isomorphic.<br />
The full result is for finite graphs of groups with virtually free vertex groups and and two-ended edge groups, but the statement is more technical -- not all such groups are QI-rigid.<br />
The main argument involves applying a new proof of Leighton's graph covering theorem.<br />
This is joint work with Sam Shepherd.<br />
<br />
===John Mackay===<br />
<br />
"Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy"<br />
<br />
The separation profile of an infinite graph was introduced by<br />
Benjamini-Schramm-Timar. It is a function which measures how<br />
well-connected the graph is by how hard it is to cut finite subgraphs<br />
into small pieces. In earlier joint work with David Hume and Romain<br />
Tessera, we introduced Poincaré profiles, generalising this concept by<br />
using p-Poincaré inequalities to measure the connected-ness of<br />
subgraphs. I will discuss this family of invariants, their applications<br />
to coarse embedding problems, and recent work finding the profiles of<br />
all connected unimodular Lie groups, where a dichotomy is exhibited.<br />
Joint with Hume and Tessera.<br />
<br />
===Benjamin Branman===<br />
<br />
"Spaces of Pants Decompositions for Surfaces of Infinite Type"<br />
<br />
We study the pants graph of surfaces of infinite type. When S is a surface of infinite type, the usual definition of the graph of pants decompositions yields a graph with infinitely many connected-components. In the first part of our talk, we study this disconnected graph. In particular, we show that the extended mapping class group of S is isomorphic to a proper subgroup of of the pants graph, in contrast to the finite-type case. In the second part of the talk, motivated by the Metaconjecture of Ivanov, we seek to endow the pants graph with additional structure. To this end, we define a coarser topology on the pants graph than the topology inherited from the graph structure. We show that our new space is path-connected, and that its automorphism group is isomorphic to the extended mapping class group.<br />
<br />
<br />
===Uri Bader===<br />
"Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity."<br />
<br />
Compact hyperbolic manifolds are very interesting geometric objects.<br />
Maybe surprisingly, they are also interesting from an algebraic point of view:<br />
They are completely determined by their fundamental groups (this is Mostow's Theorem),<br />
which could be seen as a subgroup of the integer valued invertible matrices in some dimension, GL_n(Z).<br />
When the fundamental group is the Z-points of some algebraic subgroup of GL_n we say that the manifold is arithmetic.<br />
A question arises: is there a simple geometric criterion for arithmeticity for hyperbolic manifolds?<br />
Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen.<br />
In a recent work with Fisher, Miller and Stover we proved this conjecture.<br />
Our proof is based on the theory of AREA, namely Algebraic Representation of Ergodic Actions, which I have developed with Alex Furman in recent years.<br />
In this talk I will try to survey the subject in a colloquial manner.<br />
<br />
<br />
===Omri Sarig===<br />
<br />
"(Dis)continuity of Lyapunov exponents for surface diffeomorphisms" (joint with J. Buzz and S. Crovisier)"<br />
<br />
Let f be an infinitely differentiable surface diffeomorphism. Suppose we are given a sequence of ergodic invariant measures m_n which converge weak star to an ergodic limit m. What do we need to know on m_n to guarantee that the Lyapunov exponents of m_n converge to the Lyapunov exponents of m?<br />
The main result is that if m has positive entropy, and the entropy of m_n converges to the entropy of m, then the Lyapunov exponents of m_n converge to the Lyapunov exponents of m.<br />
This is joint work with J. Buzzi and S. Crovisier.<br />
<br />
===Chris Leininger===<br />
<br />
"Billiards, symbolic coding, and cone metrics"<br />
<br />
Given a polygon in the Euclidean or hyperbolic plane a billiard trajectory in the polygon is the geodesic path of a particle in the polygon bouncing off the sides so that the angle of reflection is equal to the angle incidence. A billiard trajectory determines a symbolic coding via the sides of the polygon encountered. In this talk I will describe joint work with Erlandsson and Sadanand showing the extent to which the set of all coding sequences, the bounce spectrum, determines the shape of a hyperbolic polygon. We completely characterize those polygons which are billiard rigid (the generic case), meaning that they are determined up to isometry by their bounce spectrum. When rigidity fails for a polygon P, we parameterize the space of polygons having the same bounce spectrum at P. These results for billiards are a consequence of a rigidity/flexibility theorem for negatively curved hyperbolic cone metrics. In the talk I will explain the theorem about hyperbolic billiards, comparing/contrasting it with the Euclidean case (earlier work with Duchin, Erlandsson, and Sadanand). Then I will explain the relationship with hyperbolic cone metrics, state our rigidity/flexibility theorem for such metrics, and as time allows describe some of the ideas involved in the proofs.<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20888Dynamics Seminar 2020-20212021-02-26T02:09:25Z<p>Cwu367: /* Spring 2021 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
== Spring 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|February 3<br />
|Daniel Woodhouse (Oxford)<br />
|Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups<br />
| <br />
|-<br />
|February 10<br />
|John Mackay (Bristol)<br />
|Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy<br />
| <br />
|-<br />
|February 17<br />
|Benjamin Branman (Wisconsin)<br />
|Spaces of Pants Decompositions for Surfaces of Infinite Type<br />
| <br />
|-<br />
|February 24<br />
|Uri Bader (Weizmann Institute)<br />
|Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity.<br />
| <br />
|-<br />
|March 3<br />
|Omri Sarig (Weizmann Institute)<br />
|(Dis)continuity of Lyapunov exponents for surface diffeomorphisms (joint with J. Buzz and S. Crovisier)<br />
| <br />
|-<br />
|March 10<br />
|Chris Leininger (Rice University)<br />
|TBA<br />
| <br />
|-<br />
|March 17<br />
|Ethan Farber (Boston College)<br />
|TBA<br />
|<br />
|-<br />
|March 24<br />
|Jon Chaika (Utah)<br />
|TBA<br />
|<br />
|-<br />
|March 31<br />
|Harrison Bray (George Mason)<br />
|TBA<br />
|<br />
|-<br />
|April 7<br />
|Claire Burrin （ETH Zurich)<br />
|TBA<br />
|<br />
|-<br />
|April 21<br />
|Kasra Rafi (Toronto)<br />
|TBA<br />
|<br />
|-<br />
|April 28<br />
|Matt Bainbridge (Indiana)<br />
|TBA<br />
|}<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
===Daniel Woodhouse===<br />
<br />
"Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups"<br />
<br />
Let F be a finitely rank free group.<br />
Let w_1 and w_2 be suitable random/generic elements in F.<br />
Consider the HNN extension G = <F, t | t w_1 t^{-1} = w_2 >.<br />
It is known from existing results that G will be 1-ended and hyperbolic.<br />
We have shown that G is quasi-isometrically rigid.<br />
That is to say that if a f.g. group H is quasi-isometric to G, then G and H are virtually isomorphic.<br />
The full result is for finite graphs of groups with virtually free vertex groups and and two-ended edge groups, but the statement is more technical -- not all such groups are QI-rigid.<br />
The main argument involves applying a new proof of Leighton's graph covering theorem.<br />
This is joint work with Sam Shepherd.<br />
<br />
===John Mackay===<br />
<br />
"Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy"<br />
<br />
The separation profile of an infinite graph was introduced by<br />
Benjamini-Schramm-Timar. It is a function which measures how<br />
well-connected the graph is by how hard it is to cut finite subgraphs<br />
into small pieces. In earlier joint work with David Hume and Romain<br />
Tessera, we introduced Poincaré profiles, generalising this concept by<br />
using p-Poincaré inequalities to measure the connected-ness of<br />
subgraphs. I will discuss this family of invariants, their applications<br />
to coarse embedding problems, and recent work finding the profiles of<br />
all connected unimodular Lie groups, where a dichotomy is exhibited.<br />
Joint with Hume and Tessera.<br />
<br />
===Benjamin Branman===<br />
<br />
"Spaces of Pants Decompositions for Surfaces of Infinite Type"<br />
<br />
We study the pants graph of surfaces of infinite type. When S is a surface of infinite type, the usual definition of the graph of pants decompositions yields a graph with infinitely many connected-components. In the first part of our talk, we study this disconnected graph. In particular, we show that the extended mapping class group of S is isomorphic to a proper subgroup of of the pants graph, in contrast to the finite-type case. In the second part of the talk, motivated by the Metaconjecture of Ivanov, we seek to endow the pants graph with additional structure. To this end, we define a coarser topology on the pants graph than the topology inherited from the graph structure. We show that our new space is path-connected, and that its automorphism group is isomorphic to the extended mapping class group.<br />
<br />
<br />
===Uri Bader===<br />
"Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity."<br />
<br />
Compact hyperbolic manifolds are very interesting geometric objects.<br />
Maybe surprisingly, they are also interesting from an algebraic point of view:<br />
They are completely determined by their fundamental groups (this is Mostow's Theorem),<br />
which could be seen as a subgroup of the integer valued invertible matrices in some dimension, GL_n(Z).<br />
When the fundamental group is the Z-points of some algebraic subgroup of GL_n we say that the manifold is arithmetic.<br />
A question arises: is there a simple geometric criterion for arithmeticity for hyperbolic manifolds?<br />
Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen.<br />
In a recent work with Fisher, Miller and Stover we proved this conjecture.<br />
Our proof is based on the theory of AREA, namely Algebraic Representation of Ergodic Actions, which I have developed with Alex Furman in recent years.<br />
In this talk I will try to survey the subject in a colloquial manner.<br />
<br />
<br />
===Omri Sarig===<br />
<br />
"(Dis)continuity of Lyapunov exponents for surface diffeomorphisms" (joint with J. Buzz and S. Crovisier)"<br />
<br />
Let f be an infinitely differentiable surface diffeomorphism. Suppose we are given a sequence of ergodic invariant measures m_n which converge weak star to an ergodic limit m. What do we need to know on m_n to guarantee that the Lyapunov exponents of m_n converge to the Lyapunov exponents of m?<br />
The main result is that if m has positive entropy, and the entropy of m_n converges to the entropy of m, then the Lyapunov exponents of m_n converge to the Lyapunov exponents of m.<br />
This is joint work with J. Buzzi and S. Crovisier.<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20881Dynamics Seminar 2020-20212021-02-24T21:50:28Z<p>Cwu367: /* Spring 2021 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
== Spring 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|February 3<br />
|Daniel Woodhouse (Oxford)<br />
|Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups<br />
| <br />
|-<br />
|February 10<br />
|John Mackay (Bristol)<br />
|Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy<br />
| <br />
|-<br />
|February 17<br />
|Benjamin Branman (Wisconsin)<br />
|Spaces of Pants Decompositions for Surfaces of Infinite Type<br />
| <br />
|-<br />
|February 24<br />
|Uri Bader (Weizmann Institute)<br />
|Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity.<br />
| <br />
|-<br />
|March 3<br />
|Omri Sarig (Weizmann Institute)<br />
|(Dis)continuity of Lyapunov exponents for surface diffeomorphisms (joint with J. Buzz and S. Crovisier)<br />
| <br />
|-<br />
|March 10<br />
|Chris Leininger (Rice University)<br />
|TBA<br />
| <br />
|-<br />
|March 17<br />
|Ethan Farber (Boston College)<br />
|TBA<br />
|<br />
|-<br />
|March 24<br />
|Jon Chaika (Utah)<br />
|TBA<br />
|<br />
|-<br />
|March 31<br />
|Harrison Bray (George Mason)<br />
|TBA<br />
|<br />
|-<br />
|April 7<br />
|Claire Burrin （ETH Zurich)<br />
|TBA<br />
|<br />
|-<br />
|April 28<br />
|Matt Bainbridge (Indiana)<br />
|TBA<br />
|}<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
===Daniel Woodhouse===<br />
<br />
"Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups"<br />
<br />
Let F be a finitely rank free group.<br />
Let w_1 and w_2 be suitable random/generic elements in F.<br />
Consider the HNN extension G = <F, t | t w_1 t^{-1} = w_2 >.<br />
It is known from existing results that G will be 1-ended and hyperbolic.<br />
We have shown that G is quasi-isometrically rigid.<br />
That is to say that if a f.g. group H is quasi-isometric to G, then G and H are virtually isomorphic.<br />
The full result is for finite graphs of groups with virtually free vertex groups and and two-ended edge groups, but the statement is more technical -- not all such groups are QI-rigid.<br />
The main argument involves applying a new proof of Leighton's graph covering theorem.<br />
This is joint work with Sam Shepherd.<br />
<br />
===John Mackay===<br />
<br />
"Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy"<br />
<br />
The separation profile of an infinite graph was introduced by<br />
Benjamini-Schramm-Timar. It is a function which measures how<br />
well-connected the graph is by how hard it is to cut finite subgraphs<br />
into small pieces. In earlier joint work with David Hume and Romain<br />
Tessera, we introduced Poincaré profiles, generalising this concept by<br />
using p-Poincaré inequalities to measure the connected-ness of<br />
subgraphs. I will discuss this family of invariants, their applications<br />
to coarse embedding problems, and recent work finding the profiles of<br />
all connected unimodular Lie groups, where a dichotomy is exhibited.<br />
Joint with Hume and Tessera.<br />
<br />
===Benjamin Branman===<br />
<br />
"Spaces of Pants Decompositions for Surfaces of Infinite Type"<br />
<br />
We study the pants graph of surfaces of infinite type. When S is a surface of infinite type, the usual definition of the graph of pants decompositions yields a graph with infinitely many connected-components. In the first part of our talk, we study this disconnected graph. In particular, we show that the extended mapping class group of S is isomorphic to a proper subgroup of of the pants graph, in contrast to the finite-type case. In the second part of the talk, motivated by the Metaconjecture of Ivanov, we seek to endow the pants graph with additional structure. To this end, we define a coarser topology on the pants graph than the topology inherited from the graph structure. We show that our new space is path-connected, and that its automorphism group is isomorphic to the extended mapping class group.<br />
<br />
<br />
===Uri Bader===<br />
"Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity."<br />
<br />
Compact hyperbolic manifolds are very interesting geometric objects.<br />
Maybe surprisingly, they are also interesting from an algebraic point of view:<br />
They are completely determined by their fundamental groups (this is Mostow's Theorem),<br />
which could be seen as a subgroup of the integer valued invertible matrices in some dimension, GL_n(Z).<br />
When the fundamental group is the Z-points of some algebraic subgroup of GL_n we say that the manifold is arithmetic.<br />
A question arises: is there a simple geometric criterion for arithmeticity for hyperbolic manifolds?<br />
Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen.<br />
In a recent work with Fisher, Miller and Stover we proved this conjecture.<br />
Our proof is based on the theory of AREA, namely Algebraic Representation of Ergodic Actions, which I have developed with Alex Furman in recent years.<br />
In this talk I will try to survey the subject in a colloquial manner.<br />
<br />
<br />
===Omri Sarig===<br />
<br />
"(Dis)continuity of Lyapunov exponents for surface diffeomorphisms" (joint with J. Buzz and S. Crovisier)"<br />
<br />
Let f be an infinitely differentiable surface diffeomorphism. Suppose we are given a sequence of ergodic invariant measures m_n which converge weak star to an ergodic limit m. What do we need to know on m_n to guarantee that the Lyapunov exponents of m_n converge to the Lyapunov exponents of m?<br />
The main result is that if m has positive entropy, and the entropy of m_n converges to the entropy of m, then the Lyapunov exponents of m_n converge to the Lyapunov exponents of m.<br />
This is joint work with J. Buzzi and S. Crovisier.<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20796Dynamics Seminar 2020-20212021-02-08T17:04:58Z<p>Cwu367: /* Spring 2021 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
== Spring 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|February 3<br />
|Daniel Woodhouse (Oxford)<br />
|Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups<br />
| <br />
|-<br />
|February 10<br />
|John Mackay (Bristol)<br />
|Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy<br />
| <br />
|-<br />
|February 17<br />
|Benjamin Branman (Wisconsin)<br />
|TBA<br />
| <br />
|-<br />
|February 24<br />
|Uri Bader (Weizmann Institute)<br />
|TBA<br />
| <br />
|-<br />
|March 3<br />
|Omri Sarig (Weizmann Institute)<br />
|TBA<br />
| <br />
|-<br />
|March 10<br />
|Chris Leininger (Rice University)<br />
|TBA<br />
| <br />
|-<br />
|March 17<br />
|Ethan Farber (Boston College)<br />
|TBA<br />
|<br />
|-<br />
|March 24<br />
|Jon Chaika (Utah)<br />
|TBA<br />
|<br />
|-<br />
|March 31<br />
|Harrison Bray (George Mason)<br />
|TBA<br />
|<br />
|-<br />
|April 28<br />
|Matt Bainbridge (Indiana)<br />
|TBA<br />
|}<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
===Daniel Woodhouse===<br />
<br />
"Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups"<br />
<br />
Let F be a finitely rank free group.<br />
Let w_1 and w_2 be suitable random/generic elements in F.<br />
Consider the HNN extension G = <F, t | t w_1 t^{-1} = w_2 >.<br />
It is known from existing results that G will be 1-ended and hyperbolic.<br />
We have shown that G is quasi-isometrically rigid.<br />
That is to say that if a f.g. group H is quasi-isometric to G, then G and H are virtually isomorphic.<br />
The full result is for finite graphs of groups with virtually free vertex groups and and two-ended edge groups, but the statement is more technical -- not all such groups are QI-rigid.<br />
The main argument involves applying a new proof of Leighton's graph covering theorem.<br />
This is joint work with Sam Shepherd.<br />
<br />
===John Mackay===<br />
<br />
"Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy"<br />
<br />
The separation profile of an infinite graph was introduced by<br />
Benjamini-Schramm-Timar. It is a function which measures how<br />
well-connected the graph is by how hard it is to cut finite subgraphs<br />
into small pieces. In earlier joint work with David Hume and Romain<br />
Tessera, we introduced Poincaré profiles, generalising this concept by<br />
using p-Poincaré inequalities to measure the connected-ness of<br />
subgraphs. I will discuss this family of invariants, their applications<br />
to coarse embedding problems, and recent work finding the profiles of<br />
all connected unimodular Lie groups, where a dichotomy is exhibited.<br />
Joint with Hume and Tessera.<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20795Dynamics Seminar 2020-20212021-02-08T17:03:50Z<p>Cwu367: /* Spring Abstracts */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
== Spring 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|February 3<br />
|Daniel Woodhouse (Oxford)<br />
|Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups<br />
| <br />
|-<br />
|February 10<br />
|John Mackay (Bristol)<br />
|TBA<br />
| <br />
|-<br />
|February 17<br />
|Benjamin Branman (Wisconsin)<br />
|TBA<br />
| <br />
|-<br />
|February 24<br />
|Uri Bader (Weizmann Institute)<br />
|TBA<br />
| <br />
|-<br />
|March 3<br />
|Omri Sarig (Weizmann Institute)<br />
|TBA<br />
| <br />
|-<br />
|March 10<br />
|Chris Leininger (Rice University)<br />
|TBA<br />
| <br />
|-<br />
|March 17<br />
|Ethan Farber (Boston College)<br />
|TBA<br />
|<br />
|-<br />
|March 24<br />
|Jon Chaika (Utah)<br />
|TBA<br />
|<br />
|-<br />
|March 31<br />
|Harrison Bray (George Mason)<br />
|TBA<br />
|<br />
|-<br />
|April 28<br />
|Matt Bainbridge (Indiana)<br />
|TBA<br />
|}<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
===Daniel Woodhouse===<br />
<br />
"Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups"<br />
<br />
Let F be a finitely rank free group.<br />
Let w_1 and w_2 be suitable random/generic elements in F.<br />
Consider the HNN extension G = <F, t | t w_1 t^{-1} = w_2 >.<br />
It is known from existing results that G will be 1-ended and hyperbolic.<br />
We have shown that G is quasi-isometrically rigid.<br />
That is to say that if a f.g. group H is quasi-isometric to G, then G and H are virtually isomorphic.<br />
The full result is for finite graphs of groups with virtually free vertex groups and and two-ended edge groups, but the statement is more technical -- not all such groups are QI-rigid.<br />
The main argument involves applying a new proof of Leighton's graph covering theorem.<br />
This is joint work with Sam Shepherd.<br />
<br />
===John Mackay===<br />
<br />
"Poincaré profiles on graphs and groups, and a coarse geometric<br />
dichotomy"<br />
<br />
The separation profile of an infinite graph was introduced by<br />
Benjamini-Schramm-Timar. It is a function which measures how<br />
well-connected the graph is by how hard it is to cut finite subgraphs<br />
into small pieces. In earlier joint work with David Hume and Romain<br />
Tessera, we introduced Poincaré profiles, generalising this concept by<br />
using p-Poincaré inequalities to measure the connected-ness of<br />
subgraphs. I will discuss this family of invariants, their applications<br />
to coarse embedding problems, and recent work finding the profiles of<br />
all connected unimodular Lie groups, where a dichotomy is exhibited.<br />
Joint with Hume and Tessera.<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20712Dynamics Seminar 2020-20212021-01-31T22:12:33Z<p>Cwu367: </p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
== Spring 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|February 3<br />
|Daniel Woodhouse (Oxford)<br />
|Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups<br />
| <br />
|-<br />
|February 10<br />
|John Mackay (Bristol)<br />
|TBA<br />
| <br />
|-<br />
|February 17<br />
|Benjamin Branman (Wisconsin)<br />
|TBA<br />
| <br />
|-<br />
|February 24<br />
|Uri Bader (Weizmann Institute)<br />
|TBA<br />
| <br />
|-<br />
|March 3<br />
|Omri Sarig (Weizmann Institute)<br />
|TBA<br />
| <br />
|-<br />
|March 10<br />
|Chris Leininger (Rice University)<br />
|TBA<br />
| <br />
|-<br />
|March 17<br />
|Ethan Farber (Boston College)<br />
|TBA<br />
|<br />
|-<br />
|March 24<br />
|Jon Chaika (Utah)<br />
|TBA<br />
|<br />
|-<br />
|March 31<br />
|Harrison Bray (George Mason)<br />
|TBA<br />
|<br />
|-<br />
|April 28<br />
|Matt Bainbridge (Indiana)<br />
|TBA<br />
|}<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
===Daniel Woodhouse===<br />
<br />
"Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups"<br />
<br />
Let F be a finitely rank free group.<br />
Let w_1 and w_2 be suitable random/generic elements in F.<br />
Consider the HNN extension G = <F, t | t w_1 t^{-1} = w_2 >.<br />
It is known from existing results that G will be 1-ended and hyperbolic.<br />
We have shown that G is quasi-isometrically rigid.<br />
That is to say that if a f.g. group H is quasi-isometric to G, then G and H are virtually isomorphic.<br />
The full result is for finite graphs of groups with virtually free vertex groups and and two-ended edge groups, but the statement is more technical -- not all such groups are QI-rigid.<br />
The main argument involves applying a new proof of Leighton's graph covering theorem.<br />
This is joint work with Sam Shepherd.<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20703Dynamics Seminar 2020-20212021-01-30T06:12:43Z<p>Cwu367: /* Spring 2021 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
== Spring 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|February 3<br />
|Daniel Woodhouse (Oxford)<br />
|TBA<br />
| <br />
|-<br />
|February 10<br />
|John Mackay (Bristol)<br />
|TBA<br />
| <br />
|-<br />
|February 17<br />
|Benjamin Branman (Wisconsin)<br />
|TBA<br />
| <br />
|-<br />
|February 24<br />
|Uri Bader (Weizmann Institute)<br />
|TBA<br />
| <br />
|-<br />
|March 3<br />
|Omri Sarig (Weizmann Institute)<br />
|TBA<br />
| <br />
|-<br />
|March 10<br />
|Chris Leininger (Rice University)<br />
|TBA<br />
| <br />
|-<br />
|March 17<br />
|Ethan Farber (Boston College)<br />
|TBA<br />
|<br />
|-<br />
|March 24<br />
|Jon Chaika (Utah)<br />
|TBA<br />
|<br />
|-<br />
|March 31<br />
|Harrison Bray (George Mason)<br />
|TBA<br />
|<br />
|-<br />
|April 28<br />
|Matt Bainbridge (Indiana)<br />
|TBA<br />
|}<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20682Dynamics Seminar 2020-20212021-01-27T18:57:12Z<p>Cwu367: /* Spring 2021 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
== Spring 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|February 3<br />
|Daniel Woodhouse (Oxford)<br />
|TBA<br />
| <br />
|-<br />
|February 10<br />
|John Mackay (Bristol)<br />
|TBA<br />
| <br />
|-<br />
|February 17<br />
|Benjamin Branman (Wisconsin)<br />
|TBA<br />
| <br />
|-<br />
|February 24<br />
|Uri Bader (Weizmann Institute)<br />
|TBA<br />
| <br />
|-<br />
|March 3<br />
|Omri Sarig (Weizmann Institute)<br />
|TBA<br />
| <br />
|-<br />
|March 10<br />
|Chris Leininger (Rice University)<br />
|TBA<br />
| <br />
|-<br />
|March 17<br />
|Ethan Farber (Boston College)<br />
|TBA<br />
|<br />
|-<br />
|March 31<br />
|Harrison Bray (George Mason)<br />
|TBA<br />
|<br />
|-<br />
|April 28<br />
|Matt Bainbridge (Indiana)<br />
|TBA<br />
|}<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20675Dynamics Seminar 2020-20212021-01-27T17:18:24Z<p>Cwu367: /* Spring 2021 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
== Spring 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|February 3<br />
|Daniel Woodhouse (Oxford)<br />
|TBA<br />
| <br />
|-<br />
|February 10<br />
|John Mackay (Bristol)<br />
|TBA<br />
| <br />
|-<br />
|February 17<br />
|Benjamin Branman (Wisconsin)<br />
|TBA<br />
| <br />
|-<br />
|February 24<br />
|Uri Bader (Weizmann Institute)<br />
|TBA<br />
| <br />
|-<br />
|March 3<br />
|Omri Sarig (Weizmann Institute)<br />
|TBA<br />
| <br />
|-<br />
|March 10<br />
|Chris Leininger (Rice University)<br />
|TBA<br />
| <br />
|-<br />
|March 17<br />
|Ethan Farber (Boston College)<br />
|TBA<br />
|<br />
|-<br />
|April 28<br />
|Matt Bainbridge (Indiana)<br />
|TBA<br />
|}<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20605Dynamics Seminar 2020-20212021-01-22T13:26:47Z<p>Cwu367: /* Spring 2021 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
== Spring 2021 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|February 3<br />
|Daniel Woodhouse (Oxford)<br />
|TBA<br />
| <br />
|-<br />
|February 10<br />
|John Mackay (Bristol)<br />
|TBA<br />
| <br />
|-<br />
|February 17<br />
|Benjamin Branman (Wisconsin)<br />
|TBA<br />
| <br />
|-<br />
|February 24<br />
|Uri Bader (Weizmann Institute)<br />
|TBA<br />
| <br />
|-<br />
|March 3<br />
|Omri Sarig (Weizmann Institute)<br />
|TBA<br />
| <br />
|-<br />
|April 28<br />
|Matt Bainbridge (Indiana)<br />
|TBA<br />
|}<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20411Dynamics Seminar 2020-20212020-12-01T19:51:56Z<p>Cwu367: /* Fall Abstracts */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)<br />
<br />
===Wenyu Pan===<br />
<br />
"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"<br />
<br />
Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20410Dynamics Seminar 2020-20212020-12-01T19:50:56Z<p>Cwu367: /* Fall 2020 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps<br />
| <br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20384Dynamics Seminar 2020-20212020-11-24T21:05:39Z<p>Cwu367: /* Fall 2020 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|Limit Theorems for Quadratic Weyl Sums<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|TBA<br />
| <br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20383Dynamics Seminar 2020-20212020-11-24T21:05:06Z<p>Cwu367: /* Fall Abstracts */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|TBA<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|TBA<br />
| <br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.<br />
<br />
===Tariq Osman===<br />
<br />
"Limit Theorems for Quadratic Weyl Sums"<br />
<br />
Consider exponential sums of the form $S_N(x, \alpha) := \sum_{n = 1}^{N}e(1/2 n^2 x + n\alpha)$, known as quadratic Weyl sums. We will use homogeneous dynamics to establish a limiting distribution for $\frac{1}{\sqrt N} |S_N(x, \alpha)|$, when $\alpha$ is a fixed rational, and $x$ is chosen uniformly from the unit interval. Time permitting, we will study the tails of the limiting distribution to show that this is not the central limit theorem in disguise. (This is joint work with Francesco Cellarosi)</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20357Dynamics Seminar 2020-20212020-11-16T17:41:05Z<p>Cwu367: /* Fall Abstracts */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|TBA<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|TBA<br />
| <br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.<br />
<br />
===Nattalie Tamam===<br />
<br />
"Effective equidistribution of horospherical flows in infinite volume"<br />
<br />
Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20356Dynamics Seminar 2020-20212020-11-16T17:39:39Z<p>Cwu367: /* Fall 2020 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|Effective equidistribution of horospherical flows in infinite volume<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|TBA<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|TBA<br />
| <br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20323Dynamics Seminar 2020-20212020-11-10T18:29:02Z<p>Cwu367: /* Fall Abstracts */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|TBA<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|TBA<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|TBA<br />
| <br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.<br />
<br />
===Subhadip Dey===<br />
<br />
"Patterson-Sullivan measures for Anosov subgroups"<br />
<br />
Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the Kleinian groups and their limit sets. In this talk, we discuss an extension of this classical construction for $P$-Anosov subgroups $\Gamma$ of $G$, where $G$ is a real semisimple Lie group and $P<G$ is a parabolic subgroup. In parallel with the theory for Kleinian groups, we will discuss how one can understand the Hausdorff dimension of the limit set of $\Gamma$ in terms of a certain critical exponent. This is a joint work with Michael Kapovich.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20322Dynamics Seminar 2020-20212020-11-10T18:28:00Z<p>Cwu367: /* Fall 2020 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| <br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| <br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
|<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
|<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
|<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| <br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
| <br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|Patterson-Sullivan measures for Anosov subgroups<br />
| <br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|TBA<br />
| <br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|TBA<br />
| <br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|TBA<br />
| <br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.<br />
<br />
===Clark Butler===<br />
<br />
"Unbounded uniformizations of Grkmov hyperbolic spaces"<br />
<br />
In a fundamental work Bonk, Heinonen, and Koskela established a conformal correspondence between Gromov hyperbolic spaces and bounded uniform spaces (satisfying certain additional hypotheses) that generalized the classical conformal correspondence between the Euclidean unit disk and the hyperbolic plane. We prove a similar conformal correspondence between Gromov hyperbolic spaces and unbounded uniform spaces that extends the correspondence between the Euclidean upper half plane and the hyperbolic plane. Our primary application of this uniformization procedure is to extend a number of recent results of Bjorn-Bjorn-Shanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a Patterson-Sullivan-esque construction by realizing certain measures on these metric spaces as the boundary values of measures on uniformized Gromov hyperbolic spaces having these metric spaces as their boundaries.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20199Dynamics Seminar 2020-20212020-10-23T18:10:25Z<p>Cwu367: /* Fall 2020 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
| (local)<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
| (local)<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| (local)<br />
|-<br />
|October 28<br />
|No talk<br />
|No talk<br />
|<br />
|-<br />
|November 4<br />
|Clark Butler (Princeton)<br />
|TBA<br />
| (local)<br />
|-<br />
|November 11<br />
|Subhadip Dey (Yale)<br />
|TBA<br />
| (local)<br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|TBA<br />
| (local)<br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|TBA<br />
| (local)<br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|TBA<br />
| (local)<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20191Dynamics Seminar 2020-20212020-10-21T19:36:54Z<p>Cwu367: </p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
<br />
The zoom login info is as follows:<br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09<br />
<br />
Meeting ID: 931 6477 6780<br />
Passcode: 819612<br />
<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
| (local)<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
| (local)<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| (local)<br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|TBA<br />
| (local)<br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|TBA<br />
| (local)<br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|TBA<br />
| (local)<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20176Dynamics Seminar 2020-20212020-10-19T18:21:13Z<p>Cwu367: /* Fall 2020 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
| (local)<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
| (local)<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| (local)<br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|TBA<br />
| (local)<br />
|-<br />
|November 25<br />
|Tariq Osman (Queens)<br />
|TBA<br />
| (local)<br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|TBA<br />
| (local)<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20172Dynamics Seminar 2020-20212020-10-19T13:13:56Z<p>Cwu367: /* Fall 2020 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
| (local)<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
| (local)<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|Metrics on trees, laminations, and core entropy<br />
| (local)<br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|TBA<br />
| (local)<br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|TBA<br />
| (local)<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20171Dynamics Seminar 2020-20212020-10-19T13:13:25Z<p>Cwu367: /* Fall Abstracts */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
| (local)<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
| (local)<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|TBA<br />
| (local)<br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|TBA<br />
| (local)<br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|TBA<br />
| (local)<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).<br />
<br />
===Giulio Tiozzo===<br />
<br />
"Metrics on trees, laminations, and core entropy"<br />
<br />
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree,<br />
was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.<br />
<br />
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations<br />
defining the Julia set.<br />
<br />
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary.<br />
Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20145Dynamics Seminar 2020-20212020-10-15T18:06:04Z<p>Cwu367: /* Fall 2020 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
| (local)<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
| (local)<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|TBA<br />
| (local)<br />
|-<br />
|November 18<br />
|Nattalie Tamam (UCSD)<br />
|TBA<br />
| (local)<br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|TBA<br />
| (local)<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20136Dynamics Seminar 2020-20212020-10-14T17:00:00Z<p>Cwu367: /* Fall 2020 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
| (local)<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
| (local)<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|TBA<br />
| (local)<br />
|-<br />
|December 2<br />
|Wenyu Pan (Chicago)<br />
|TBA<br />
| (local)<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20134Dynamics Seminar 2020-20212020-10-14T14:21:42Z<p>Cwu367: /* Fall 2020 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey (Boston College)<br />
|Slices of Thurston's Master Teapot<br />
| (local)<br />
|-<br />
|October 14<br />
|Daniel Thompson (Ohio State)<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
| (local)<br />
|-<br />
|October 21<br />
|Giulio Tiozzo (Toronto)<br />
|TBA<br />
| (local)<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20133Dynamics Seminar 2020-20212020-10-14T14:20:14Z<p>Cwu367: /* Fall 2020 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey<br />
|Slices of Thurston's Master Teapot<br />
| (Boston College)<br />
|-<br />
|October 14<br />
|Daniel Thompson<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
| (Ohio State)<br />
|-<br />
|October 21<br />
|Giulio Tiozzo<br />
|TBA<br />
| (Toronto)<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20129Dynamics Seminar 2020-20212020-10-13T20:29:40Z<p>Cwu367: /* Fall 2020 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey<br />
|Slices of Thurston's Master Teapot<br />
| (Boston College)<br />
|-<br />
|October 14<br />
|Daniel Thompson<br />
|Strong ergodic properties for equilibrium states in non-positive curvature<br />
| (Ohio State)<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20128Dynamics Seminar 2020-20212020-10-13T20:29:05Z<p>Cwu367: /* Fall Abstracts */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey<br />
|Slices of Thurston's Master Teapot<br />
| (Boston College)<br />
|-<br />
|October 14<br />
|Daniel Thompson<br />
|TBA<br />
| (Ohio State)<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.<br />
<br />
<br />
===Daniel Thompson===<br />
<br />
"Strong ergodic properties for equilibrium states in non-positive curvature"<br />
<br />
Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang).</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20057Dynamics Seminar 2020-20212020-10-01T21:37:36Z<p>Cwu367: /* Kathryn Lindsey */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey<br />
|Slices of Thurston's Master Teapot<br />
| (Boston College)<br />
|-<br />
|October 14<br />
|Daniel Thompson<br />
|TBA<br />
| (Ohio State)<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20056Dynamics Seminar 2020-20212020-10-01T21:36:35Z<p>Cwu367: /* Fall Abstracts */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey<br />
|Slices of Thurston's Master Teapot<br />
| (Boston College)<br />
|-<br />
|October 14<br />
|Daniel Thompson<br />
|TBA<br />
| (Ohio State)<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]<br />
<br />
<br />
===Kathryn Lindsey===<br />
<br />
"Slices of Thurston's Master Teapot"<br />
<br />
Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a ``restricted iterated function system.'' An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20055Dynamics Seminar 2020-20212020-10-01T21:34:56Z<p>Cwu367: /* Fall 2020 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey<br />
|Slices of Thurston's Master Teapot<br />
| (Boston College)<br />
|-<br />
|October 14<br />
|Daniel Thompson<br />
|TBA<br />
| (Ohio State)<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20042Dynamics Seminar 2020-20212020-09-30T20:31:32Z<p>Cwu367: /* Chenxi Wu */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey<br />
|TBA<br />
| (Boston College)<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim. [https://wuchenxi.github.io/graph.pdf Slides]</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20041Dynamics Seminar 2020-20212020-09-30T20:30:45Z<p>Cwu367: /* Fall 2020 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptoic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey<br />
|TBA<br />
| (Boston College)<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20031Dynamics Seminar 2020-20212020-09-29T19:27:19Z<p>Cwu367: /* Chenxi Wu */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptomatic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey<br />
|TBA<br />
| (Boston College)<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on curve complexes and free factor complexes"<br />
<br />
The curve complex of a closed surface is a simplicial complex where the vertices are simple closed curves up to isotopy and faces are curves that are disjoint, and an analogy for the curve complex in the setting of Out(F_n) is the free factor complex. A pseudo-Anosov map induces a map from the curve graph to itself, and a basic question is to study the asymptotic translation length which is known to be a non-zero rational number. I will review some prior results on the study of this asymptotic translation length, as well as some of their analogies in the setting of free factor complexes. The latter part is an ongoing project with Hyrungryul Baik and Dongryul Kim.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=20030Dynamics Seminar 2020-20212020-09-29T19:24:43Z<p>Cwu367: /* Fall 2020 */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptomatic translation lengths on curve complexes and free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey<br />
|TBA<br />
| (Boston College)<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on free factor complex"<br />
<br />
The free factor complex is a graph theoretic analogy of the curve complex of surfaces. I will review some results about the asymptotic translation length of pseudo-Anosov action on curve complexes, in particular, the estimate of asymptotic translation lengths for pseudo-Anosovs with the same mapping torus, and the analogous results in the setting of free factor complexes. This will include prior works with Hyungryul Baik, Eiko Kin and Hyunshik Shin, as well as ongoing work with Hyungryul Baik and Dongryul Kim.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=19947Dynamics Seminar 2020-20212020-09-24T17:03:58Z<p>Cwu367: /* Chenxi Wu */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptomatic translation lengths on free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey<br />
|TBA<br />
| (Boston College)<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
"Asymptotic translation lengths on free factor complex"<br />
<br />
The free factor complex is a graph theoretic analogy of the curve complex of surfaces. I will review some results about the asymptotic translation length of pseudo-Anosov action on curve complexes, in particular, the estimate of asymptotic translation lengths for pseudo-Anosovs with the same mapping torus, and the analogous results in the setting of free factor complexes. This will include prior works with Hyungryul Baik, Eiko Kin and Hyunshik Shin, as well as ongoing work with Hyungryul Baik and Dongryul Kim.</div>Cwu367https://wiki.math.wisc.edu/index.php?title=Dynamics_Seminar_2020-2021&diff=19946Dynamics Seminar 2020-20212020-09-24T16:54:38Z<p>Cwu367: /* Chenxi Wu */</p>
<hr />
<div>The [[Dynamics Seminar]] meets virtually on '''Wednesdays''' from '''2:30pm - 3:20pm'''.<br />
<br> <br />
For more information, contact Chenxi Wu.<br />
To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Fall 2020 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 16<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations I<br />
| (local)<br />
|-<br />
|September 23<br />
|Andrew Zimmer (Wisconsin)<br />
|An introduction to Anosov representations II<br />
| (local)<br />
|-<br />
|September 30<br />
|Chenxi Wu (Wisconsin)<br />
|Asymptomatic translation lengths on free factor complexes<br />
| (local)<br />
|-<br />
|October 7<br />
|Kathryn Lindsey<br />
|TBA<br />
| (Boston College)<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
===Andrew Zimmer===<br />
<br />
"An introduction to Anosov representations"<br />
<br />
Anosov representations are a special class of representations of finitely generated groups into Lie groups, which are defined using ideas from dynamics (namely, the theory of Anosov flows). In this talk, I will explain the definition (in a special case), give some examples, and describe some properties. I will focus on the case of representations into the general linear group where no background knowledge about Lie groups is required.<br />
<br />
<br />
===Chenxi Wu===<br />
<br />
Asymptotic translation lengths on free factor complex<br />
<br />
Abstract:</div>Cwu367