# Ergodic Theory and Dynamical Systems Seminar

##### The seminars are held on Tuesdays at 14:00.
Organizers: Joel Moreira
##### Term 2
• January 19, 2021
Scott Schmieding (University of Denver)

Title: Local P entropy and stabilized automorphism groups.
Abstract: For a homeomorphism of a compact metric space T : X → X, the stabilized automorphism group of this system consists of all self-homeomorphisms of X which commute with some power of T. Motivated by studying this stabilized group in the setting of symbolic systems, we will introduce and discuss a certain entropy for groups called local P entropy. We show that local P entropy can be used to give a complete classification of the stabilized automorphisms groups of full shifts; in particular, we show the stabilized groups for the 2-shift and the 3-shift are not isomorphic.

• February 2, 2021
Tom Ward (Leeds)

Title: Time-changes preserving zeta functions
Abstract: A dynamical system with finitely many periodic orbits of each period has an associated collection of possible time-changes that preserve the property of counting periodic points for some system. Intersecting over all dynamical systems gives a monoid of time-changes that have this property for all such systems. This universal monoid has remarkable properties: the only polynomials it contains are the monomials, and it is uncountable.

• February 9, 2021
Amir Algom (Penn State)

Title: On the decay of the Fourier transform of self-conformal measures
Abstract: Let P be a self-conformal measure with respect to an IFS consisting of finitely many smooth contractions of [0,1]. Assuming a mild and natural condition on the derivative cocycle, we prove that the Fourier transform of P decays to zero at infinity. This is related to the highly active study of the properties of the Fourier transform of dynamically defined measures, dating back to the important work of Erdos about Bernoulli convolutions in the late 1930's. This is a joint work with Federico Rodriguez Hertz and Zhiren Wang.

• February 16, 2021
Andrey Gogolyev (Ohio State)

Title: Rigidity for higher dimensional expanding maps
Abstract: Expanding maps are self covers of smooth compact manifolds which expand the lengths of all non-zero tangent vectors. Classification of such maps up to topological conjugacy is known due to work of Shub, Franks and Gromov. Classification up to smooth conjugacy should be quite different because periodic points of expanding maps carry invariants of C^1 conjugacy. Shub and Sullivan classified expanding maps on the circle up to smooth conjugacy on the circle. I will explain smooth classification of expanding maps in higher dimensions on and open dense set in the space of expanding maps. Joint work with F. Rodriguez Hertz.

• February 23, 2021
Anh Le (Ohio State University)

Title: Sets of multiplicative recurrence and applications
Abstract: A set R of positive rationals is a set of multiplicative recurrence if for any finite coloring of $\mathbb N$, there are monochromatic x, y such that x/y is in R. In this talk, we will give criteria for when sets of the form $\{(an+b)/(cn+d): n \in \mathbb N\}$ are sets of multiplicative recurrence. Consequently, we recover two recent results in number theory regarding completely multiplicative functions and the Omega function. This is based on a joint work with S. Donoso, J. Moreira and W. Sun.

• March 16, 2021. Time: 4pm (Note the unusual time!)
Lewis Bowen (U. Texas, Austin)

Title: A New Infinite-Dimensional Multiplicative Ergodic Theorem
Abstract: In 1960, Furstenberg and Kesten introduced the problem of describing the asymptotic behavior of products of random matrices as the number of factors tends to infinity. Oseledets’ proved that such products, after normalization, converge almost surely. This theorem has wide-ranging applications to smooth ergodic theory and rigidity theory. It has been generalized to products of random operators on Banach spaces by Ruelle and others. I will explain a new infinite-dimensional generalization based on von Neumann algebra theory which accommodates continuous Lyapunov distribution. This will be a gentle introductory-style talk; no knowledge of von Neumann algebras will be assumed. This is joint work with Ben Hayes (U. Virginia) and Yuqing Frank Lin (Ben-Gurion U.).

##### Term 1
• October 20, 2020
John Griesmer (Colorado School of Mines) Slides

Title: Single recurrence: recent results, open problems, and mysterious examples
Abstract: The Poincaré Recurrence Theorem says that whenever $(X,\mu,T)$ is a probability measure preserving system and $A\subset X$ has positive measure, there is an $n\in \mathbb N$ such that $\mu(A\cap T^{-n}A)>0$. H. Furstenberg and A. Sárközy independently proved an appealing refinement: under the same hypotheses, one can conclude that $\mu(A\cap T^{-n^2}A)>0$ for some $n\in \mathbb N$. One may ask, in general, for which sets $S\subset \mathbb N$ can one conclude that $\mu(A\cap T^{-n}A)>0$ for some $n\in S$? This is the study of single recurrence in measurable dynamics. Many examples and non-examples are known, but a satisfying description remains elusive. This talk will summarize the current state of knowledge regarding single recurrence, advertise well-known open problems, and introduce some more tractable variations. We will also provide explicit examples of sets whose recurrence properties are unknown.

• October 27, 2020
Anish Ghosh (Tata Institute of Fundamental Research)

Title: Quadratic forms and ergodic theory
Abstract: I will discuss connections between ergodic theory on homogeneous spaces and the study of quadratic forms. This connection began with Margulis's famous resolution of Oppenheim's conjecture. I will talk about some recent work which studies effective versions of Margulis's theorem, especially in the context of inhomogeneous quadratic forms.

• November 3, 2020

Title: Closed Geodesics on Surfaces without Conjugate Points
Abstract: We obtain Margulis-type asymptotic estimates for the number of free homotopy classes of closed geodesics on certain manifolds without conjugate points. Our results cover all compact surfaces of genus at least 2 without conjugate points. This is based on a join work with Vaughn Climenhaga and Gerhard Knieper.

• November 10, 2020
Brian Chung (U Chicago)

Title: Stationary measure and orbit closure classification for random walks on surfaces
Abstract: We study the problem of classifying stationary measures and orbit closures for non-abelian action on surfaces. Using a result of Brown and Rodriguez Hertz, we show that under a certain average growth condition, the orbit closures are either finite or dense. Moreover, every infinite orbit equidistributes on the surface. This is analogous to the results of Benoist-Quint and Eskin-Lindenstrauss in the homogeneous setting, and the result of Eskin-Mirzakhani in the setting of moduli spaces of translation surfaces. We then consider the problem of verifying this growth condition in concrete settings. In particular, we apply the theorem to two settings, namely discrete perturbations of the standard map and the Out(F2)-action on a certain character variety. We verify the growth condition analytically in the former setting, and verify numerically in the latter setting.

• November 17, 2020
Osama Khalil (Utah)

Title: Random Walks, Spectral Gaps, and Khinchine Theorem on Fractals.
Abstract: In 1984, Mahler asked how well typical points on Cantor’s set can be approximated by rational numbers. His question fits within a program, set out by himself in the 1930s, attempting to determine conditions under which subsets of $\mathbb{R}^n$ inherit the Diophantine properties of the ambient space. Since approximability of typical points in Euclidean space by rational points is governed by Khinchine’s classical theorem, the ultimate form of Mahler’s question asks whether an analogous zero-one law holds for fractal measures. Significant progress has been achieved in recent years, albeit, almost all known results have been of “convergence type”.

In this talk, we will discuss the first instances where a complete analogue of Khinchine’s theorem for fractal measures is obtained. The main new ingredient is an effective equidistribution theorem for certain fractal measures on the space of unimodular lattices. The latter is established via a new technique involving the construction of S-arithmetic Markov operators possessing a spectral gap and encoding the arithmetic structure of the maps generating the fractal. This is joint work with Manuel Luethi.
• December 1, 2020
Pankaj Vishe (Durham)

Title: A sparse equidistribution result for $(SL(2,\mathbb{R})/\Gamma_0)^n$
Abstract: Let $G=(SL(2,\mathbb{R})^n$ and $\Gamma=\Gamma_0^n$ where $\Gamma_0$ is a co-compact lattice in $SL(2,\mathbb{R})$. Let $U(x_1,...,x_n)$ denote a standard n-parameter unipotent expanding horospherical subgroup in G. The equidistribution of whole U orbits follow from Ratner's equidistribution theorems. It is believed that sparse arithmetic subsets of these orbits should equidistribute for every orbit, however, results of these type have been extremely rare. Typically one is able to obtain results of metric nature. We prove that the sparse subsets of all U orbits evaluated at integer times which lie on a quadric hypersurface equidistribute in $G/\Gamma$ as long as $n\geq 481$. Key tools here are a combination of bounds for twisted averages of functions over horocycle flow and the Hardy-Littlewood circle method.

• December 8, 2020

Title: Introduction to tiling billiards
Abstract: I will tell you about the family of dynamical systems I am passionate about for almost four years now. This family models, in a naive way, a movement of refracted light in a tiling of the plane. The light follows Snell’s law with refraction coefficient equal to -1. This may seem odd as a choice of refraction coefficient... but this definition leads to rich mathematics already in the simplest tilings. In practice, one deals with families of interval exchange transformations with flips and continued fraction algorithms. In the talk I will concentrate myself on the case of periodic triangle tiling, the one that I understand the best. Recently another interpretation of tiling billiards as a physical system has been discovered. The tiling billiard trajectories model, for a large class of tilings, the movement of electrons on the Fermi surface of a metal in the presence of a magnetic field! (a so-called Novikov’s problem) The talk is introductory and elementary. The best way to prepare for it is to watch a short animated movie by Ofir David called Refraction tilings. One image is worth a million words.

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