Seminar 10.22.20 Best

Title: The Furstenberg-Sárközy theorem and asymptotic total ergodicity

Speaker: Andrew Best – Ohio State University

Abstract: The Furstenberg-Sárközy theorem asserts that the difference set E-E of a subset E of the natural numbers with positive upper density contains a (nonzero) square. Furstenberg’s approach relies on a correspondence principle and a version of the Poincaré recurrence theorem along squares; the latter is shown via the result that for any measure-preserving system $(X,\mathcal{B},\mu,T)$ and set A with positive measure, the ergodic average $\frac{1}{N} \sum_{n=1}^N \mu(A \cap T^{-n^2}A)$ has a positive limit c(A) as N tends to infinity. Motivated — by what? we shall see — to optimize the value of c(A), we define the notion of asymptotic total ergodicity in the setting of modular rings $\mathbb{Z}/N\mathbb{Z}$. We show that a sequence of modular rings (Z/N_m Z) is asymptotically totally ergodic if and only if the least prime factor of N_m grows to infinity. From this fact, we derive some combinatorial consequences. These results are based on joint work with Vitaly Bergelson.

Zoom recording available here

Seminar 10.16.20 Wang

Title: Central Limit Theorem for equilibrium measures in dynamical systems

Speaker: Tianyu Wang – Ohio State University

Abstract: Central limit theorem of certain class of equilibrium measures is a heavily studied statistical property in smooth dynamics. In the first half of the talk, I will briefly introduce some common strategies to study CLT that are useful in many classic settings, e.g. Anosov flows, expanding maps on the unit circle, (countable) Markov shift, etc. In the second half, I will show how specification can be applied to derive an asymptotic version of CLT for the equilibrium measures in the case of geodesic flow on non-positively curved rank-one manifold. This method is first introduced by Denker, Senti, Zhang and the result is based on a recent joint work with Dan Thompson.

Zoom recording available here

Seminar 10.9.20 Demers

Title: Thermodynamic Formalism for Sinai Billiards

Speaker: Mark Demers – Fairfield University

Abstract: While the ergodic properties of Sinai billiards with respect to the SRB measure are well understood, there have been few studies of other invariant measures and equilibrium states. As a step in this direction, we study the family of geometric potentials $– t \log (J^uT)$, $t>0$. For any finite horizon Sinai billiard map $T$, we find $t_* >1$ such that for each $t \in (0, t_*)$, there exists a unique equilibrium state $\mu_t$ for the potential. We show that $\mu_t$ is exponentially mixing for H\”older observables, and that the pressure function $P(t)$ is analytic on $(0,t_*)$. This extends our recent results for the case $t=0$, corresponding to the measure of maximal entropy. This is joint work with Viviane Baladi.

Zoom recording available here

Pdf of slides available here

Seminar 10.2.20 Das

Title: Successive minima of lattice trajectories and topological games to compute fractal dimensions

Speaker: Tushar Das – University of Wisconsin – La Crosse

Abstract: We present certain sketches of a program, developed in collaboration with Lior Fishman, David Simmons, and Mariusz Urbanski, which extends the parametric geometry of numbers (initiated by Schmidt and Summerer, and deepened by Roy) to Diophantine approximation for systems of m linear forms in n variables, and establishes a new connection to the metric theory via a variational principle that computes the fractal dimensions of various sets of number-theoretic interest. Applications of our results include computing the Hausdorff and packing dimensions of the set of singular systems of linear forms and showing they are equal, resolving a conjecture of Kadyrov, Kleinbock, Lindenstrauss and Margulis, as well as a question of Bugeaud, Cheung and Chevallier. As a corollary of the correspondence principle (initiated by Dani, and deepened by Kleinbock and Margulis), the divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs has equal Hausdorff and packing dimensions. Highlights of our program include the introduction of certain combinatorial objects that we call templates, which arise from a dynamical study of Minkowski’s successive minima in the geometry of numbers; as well as a new variant of Schmidt’s game designed to compute the Hausdorff and packing dimensions of any set in a doubling metric space. The talk will be accessible to students and faculty interested in some convex combination of homogeneous dynamics, Diophantine approximation and fractal geometry. I hope to present a sampling of open questions and directions that have yet to be explored, some of which may be pursued by either following or adapting the technology described in my talk.

Zoom recording available here

Pdf of slides available here