# Difference between revisions of "Dynamics Seminar 2020-2021"

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

## Revision as of 13:51, 1 December 2020

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

The zoom login info is as follows:

Join Zoom Meeting https://uwmadison.zoom.us/j/93164776780?pwd=anE2Y3RhWk1VR0lDa0hnMzhPTTJEUT09

Meeting ID: 931 6477 6780 Passcode: 819612

## Contents

## 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 |

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