Dynamics Seminar 2020-2021

The Dynamics Seminar meets virtually on Wednesdays from 2:30pm - 3:20pm.
For more information, contact Chenxi Wu. To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu

Meetings are on Zoom. To get Zoom info email Chenxi Wu.

Spring 2021

date	speaker	title	host(s)
February 3	Daniel Woodhouse (Oxford)	Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups
February 10	John Mackay (Bristol)	Poincaré profiles on graphs and groups, and a coarse geometric dichotomy
February 17	Benjamin Branman (Wisconsin)	Spaces of Pants Decompositions for Surfaces of Infinite Type
February 24	Uri Bader (Weizmann Institute)	Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity.
March 3	Omri Sarig (Weizmann Institute)	(Dis)continuity of Lyapunov exponents for surface diffeomorphisms (joint with J. Buzz and S. Crovisier)
March 10	Chris Leininger (Rice University)	Billiards, symbolic coding, and cone metrics
March 17	Ethan Farber (Boston College)	Constructing pseudo-Anosovs from expanding interval maps
March 24	Jon Chaika (Utah)	A strange limit of horocycle ergodic measures in a stratum of translation surfaces
March 31	Harrison Bray (George Mason)	Volume-entropy rigidity for convex real projective manifolds
April 7	Claire Burrin （ETH Zurich)	A sparse equidistribution problem for expanding horocycles on the modular surface
April 21	Kasra Rafi (Toronto)	Absolutely continuous stationary measures for the mapping class group
April 28	Matt Bainbridge (Indiana)	Haupt's theorem for strata of holomorphic one-forms and isoperiodic foliations

Fall 2020

date	speaker	title	host(s)
September 16	Andrew Zimmer (Wisconsin)	An introduction to Anosov representations I
September 23	Andrew Zimmer (Wisconsin)	An introduction to Anosov representations II
September 30	Chenxi Wu (Wisconsin)	Asymptoic translation lengths on curve complexes and free factor complexes
October 7	Kathryn Lindsey (Boston College)	Slices of Thurston's Master Teapot
October 14	Daniel Thompson (Ohio State)	Strong ergodic properties for equilibrium states in non-positive curvature
October 21	Giulio Tiozzo (Toronto)	Metrics on trees, laminations, and core entropy
October 28	No talk	No talk
November 4	Clark Butler (Princeton)	"Unbounded uniformizations of Grkmov hyperbolic spaces"
November 11	Subhadip Dey (Yale)	Patterson-Sullivan measures for Anosov subgroups
November 18	Nattalie Tamam (UCSD)	Effective equidistribution of horospherical flows in infinite volume
November 25	Tariq Osman (Queens)	Limit Theorems for Quadratic Weyl Sums
December 2	Wenyu Pan (Chicago)	Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps

Spring Abstracts

Daniel Woodhouse

"Quasi-isometric Rigidity of graphs of free groups with cyclic edge groups"

Let F be a finitely rank free group. Let w_1 and w_2 be suitable random/generic elements in F. Consider the HNN extension G = <F, t | t w_1 t^{-1} = w_2 >. It is known from existing results that G will be 1-ended and hyperbolic. We have shown that G is quasi-isometrically rigid. That is to say that if a f.g. group H is quasi-isometric to G, then G and H are virtually isomorphic. 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. The main argument involves applying a new proof of Leighton's graph covering theorem. This is joint work with Sam Shepherd.

John Mackay

"Poincaré profiles on graphs and groups, and a coarse geometric dichotomy"

The separation profile of an infinite graph was introduced by Benjamini-Schramm-Timar. It is a function which measures how well-connected the graph is by how hard it is to cut finite subgraphs into small pieces. In earlier joint work with David Hume and Romain Tessera, we introduced Poincaré profiles, generalising this concept by using p-Poincaré inequalities to measure the connected-ness of subgraphs. I will discuss this family of invariants, their applications to coarse embedding problems, and recent work finding the profiles of all connected unimodular Lie groups, where a dichotomy is exhibited. Joint with Hume and Tessera.

Benjamin Branman

"Spaces of Pants Decompositions for Surfaces of Infinite Type"

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.

Uri Bader

"Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity."

Compact hyperbolic manifolds are very interesting geometric objects. Maybe surprisingly, they are also interesting from an algebraic point of view: They are completely determined by their fundamental groups (this is Mostow's Theorem), which could be seen as a subgroup of the integer valued invertible matrices in some dimension, GL_n(Z). When the fundamental group is the Z-points of some algebraic subgroup of GL_n we say that the manifold is arithmetic. A question arises: is there a simple geometric criterion for arithmeticity for hyperbolic manifolds? Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen. In a recent work with Fisher, Miller and Stover we proved this conjecture. 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. In this talk I will try to survey the subject in a colloquial manner.

Omri Sarig

"(Dis)continuity of Lyapunov exponents for surface diffeomorphisms" (joint with J. Buzz and S. Crovisier)"

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? 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. This is joint work with J. Buzzi and S. Crovisier.

Chris Leininger

"Billiards, symbolic coding, and cone metrics"

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.

Ethan Farber

"Constructing pseudo-Anosovs from expanding interval maps"

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.

Jon Chaika

"A strange limit of horocycle ergodic measures in a stratum of translation surfaces"

The main result of this talk is that in the space of unit area translation surfaces with one cone point there is a weak-star limit of measures on periodic horocycles that is fully supported in the 7-dimensional space but gives positive measure to a 3-dimensional submanifold. As a consequence we obtain a non-genericity result for the horocycle flow in this space. I will define the terminology. This is joint work with Osama Khalil and John Smillie.

Harrison Bray

"Volume-entropy rigidity for convex real projective manifolds"

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.

Claire Burrin

"A sparse equidistribution problem for expanding horocycles on the modular surface"

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.

Kasra Rafi

"Absolutely continuous stationary measures for the mapping class group"

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.

Matt Bainbridge

"Haupt's theorem for strata of holomorphic one-forms and isoperiodic foliations"

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.

Fall Abstracts

Andrew Zimmer

"An introduction to Anosov representations"

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.

Chenxi Wu

"Asymptotic translation lengths on curve complexes and free factor complexes"

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. Slides

Kathryn Lindsey

"Slices of Thurston's Master Teapot"

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.

Daniel Thompson

"Strong ergodic properties for equilibrium states in non-positive curvature"

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).

Giulio Tiozzo

"Metrics on trees, laminations, and core entropy"

The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree, 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.

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 defining the Julia set.

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. Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.

Clark Butler

"Unbounded uniformizations of Grkmov hyperbolic spaces"

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.

Subhadip Dey

"Patterson-Sullivan measures for Anosov subgroups"

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.

Nattalie Tamam

"Effective equidistribution of horospherical flows in infinite volume"

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.

Tariq Osman

"Limit Theorems for Quadratic Weyl Sums"

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)

Wenyu Pan

"Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps"

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 $\Gamma\backslash \mathbb{H}^n$.