Difference between revisions of "Dynamics Seminar 20202021"
(→Spring Abstracts) 

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To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@ggroups.wisc.edu  To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@ggroups.wisc.edu  
−  +  Meetings are on Zoom. To get Zoom info email Chenxi Wu.  
−  
−  
−  
−  
−  
Line 30:  Line 25:  
February 10  February 10  
John Mackay (Bristol)  John Mackay (Bristol)  
−    +  Poincaré profiles on graphs and groups, and a coarse geometric 
+  dichotomy  
    
    
February 17  February 17  
Benjamin Branman (Wisconsin)  Benjamin Branman (Wisconsin)  
−    +  Spaces of Pants Decompositions for Surfaces of Infinite Type 
    
    
February 24  February 24  
Uri Bader (Weizmann Institute)  Uri Bader (Weizmann Institute)  
−    +  Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity. 
    
    
March 3  March 3  
Omri Sarig (Weizmann Institute)  Omri Sarig (Weizmann Institute)  
−    +  (Dis)continuity of Lyapunov exponents for surface diffeomorphisms (joint with J. Buzz and S. Crovisier) 
    
    
March 10  March 10  
Chris Leininger (Rice University)  Chris Leininger (Rice University)  
−    +  Billiards, symbolic coding, and cone metrics 
    
    
March 17  March 17  
Ethan Farber (Boston College)  Ethan Farber (Boston College)  
−    +  Constructing pseudoAnosovs from expanding interval maps 
    
    
March 24  March 24  
Jon Chaika (Utah)  Jon Chaika (Utah)  
−    +  A strange limit of horocycle ergodic measures in a stratum of 
+  translation surfaces  
    
    
March 31  March 31  
Harrison Bray (George Mason)  Harrison Bray (George Mason)  
+  Volumeentropy 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)  
TBA  TBA  
    
Line 174:  Line 181:  
all connected unimodular Lie groups, where a dichotomy is exhibited.  all connected unimodular Lie groups, where a dichotomy is exhibited.  
Joint with Hume and Tessera.  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 connectedcomponents. 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 finitetype 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 pathconnected, 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 Zpoints 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 pseudoAnosovs from expanding interval maps"  
+  
+  The celebrated NielsenThurston 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) pseudoAnosov. Of these three types, pseudoAnosovs are the most intriguing to dynamicists, with connections to symbolic dynamics and flat geometry. In this talk we investigate a construction of generalized pseudoAnosovs 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 pseudoAnosov, 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 weakstar limit of  
+  measures on periodic horocycles that is fully supported in the  
+  7dimensional space but gives positive measure to a 3dimensional  
+  submanifold. As a consequence we obtain a nongenericity 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===  
+  
+  "Volumeentropy rigidity for convex real projective manifolds"  
+  
+  I will discuss joint work with Constantine, building on joint work with Adeboye and Constantine, on a volumeentropy rigidity result for finite volume strictly convex projective manifolds in dimension at least 3. The result is a BessonCourtoisGallot 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.  
+  
== Fall Abstracts ==  == Fall Abstracts == 
Latest revision as of 12:53, 6 April 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@ggroups.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)  Quasiisometric 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 pseudoAnosovs 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)  Volumeentropy 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)  TBA  
April 28  Matt Bainbridge (Indiana)  TBA 
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 nonpositive 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)  PattersonSullivan 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
"Quasiisometric 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 1ended and hyperbolic. We have shown that G is quasiisometrically rigid. That is to say that if a f.g. group H is quasiisometric 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 twoended edge groups, but the statement is more technical  not all such groups are QIrigid. 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 BenjaminiSchrammTimar. It is a function which measures how wellconnected 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 pPoincaré inequalities to measure the connectedness 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 connectedcomponents. 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 finitetype 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 pathconnected, 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 Zpoints 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 pseudoAnosovs from expanding interval maps"
The celebrated NielsenThurston 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) pseudoAnosov. Of these three types, pseudoAnosovs are the most intriguing to dynamicists, with connections to symbolic dynamics and flat geometry. In this talk we investigate a construction of generalized pseudoAnosovs 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 pseudoAnosov, 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 weakstar limit of measures on periodic horocycles that is fully supported in the 7dimensional space but gives positive measure to a 3dimensional submanifold. As a consequence we obtain a nongenericity 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
"Volumeentropy rigidity for convex real projective manifolds"
I will discuss joint work with Constantine, building on joint work with Adeboye and Constantine, on a volumeentropy rigidity result for finite volume strictly convex projective manifolds in dimension at least 3. The result is a BessonCourtoisGallot 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.
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 pseudoAnosov 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 nonzero 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 selfmap 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 nonpositive 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 piecewiselinear 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 BjornBjornShanmugalingam for Besov spaces on compact metric spaces to Besov spaces on proper metric spaces. These results are derived through a PattersonSullivanesque 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
"PattersonSullivan measures for Anosov subgroups"
PattersonSullivan 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 BowenMargulisSullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyattype 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 resonancefree region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.