# Seminar: Ergodic Theory and Dynamcial Systems

Time: 4:00 pm (unless otherwise noted)

Current schedule for the semester (spring 2023)

March 8: Younghwan Son (POSTECH)

March 22: Haritha Cheriyath (Tata Institute)

April 5: Andrew Best (BIMSA, China)

April 19: Omri Sarig (Weizmann Institute)

May 10: Jiyoung Han (KIAS)

May 17: Florian Richter (EPFL)

3. 8: Younghwan Son

Title: Joint normality of representations of numbers

Abstract: In this talk we deal with the phenomenon of joint ergodicity of piecewise monotone maps on the unit interval. Especially we present a result on pointwise joint ergodicity of the times b map and the Gauss map, which implies that almost every numbers are jointly normal with respect to b-expansion and continued fraction expansion.

This is a joint work with Vitaly Bergelson.

3. 22: Haritha Cheriyath

Title: OPEN DYNAMICS ON SUBSHIFTS OF FINITE TYPE

Abstract: Dynamical systems can be broadly classified into closed and open systems. In a (traditional) closed system, the orbit of a point lies in the state space for all time, whereas in an open system, the orbit of a point may eventually escape from the state space through a hole. The notion of open dynamical systems was introduced by Pianigiani and Yorke in 1979, motivated by the dynamics of a ball on a billiard table with pockets. It has attracted the attention of researchers since then especially due to its wide applications.

In this talk, we consider an irreducible subshift of finite type and study the average rate at which the orbits escape into the hole (termed as the escape rate). This problem turns out to be an interesting application of a combinatorial question of counting the number of words of given length not containing any of the words from a fixed collection as subwords. Using this, we compare the escape rates into different holes. We also present some applications of our results in computing the Perron eigenvalues and eigenvectors of any non-negative integer matrix and obtaining a combinatorial expression for the unique measure of maximal entropy on a subshift of finite type.

4. 5 Andrew Best

Title: On the polynomial Szemerédi theorem over finite commutative rings

Abstract: The polynomial Szemerédi theorem implies that, for any $\delta \in (0,1)$, any family $\{P_1,\ldots, P_m\} \subset \Z[y]$ of nonconstant polynomials with constant term zero, and any sufficiently large $N \in \N$, every subset of $\{1,\ldots, N\}$ of cardinality at least $\delta N$ contains a nontrivial configuration of the form $\{x,x+P_1(y),\ldots, x+P_m(y)\}$. When the polynomials are assumed independent, one can expect a sharper result to hold over finite fields, special cases of which were proven recently in various articles by Bourgain and others, culminating with a 2018 result of Peluse, which deals with the general case of independent polynomials. In this talk we discuss, over general finite commutative rings, a version of the polynomial Szemerédi theorem for multivariable independent polynomials $\{P_1,\ldots, P_m\} \subset \Z[y_1,\ldots, y_n]$, deriving new combinatorial consequences, such as the following. Let $\mathcal R$ be a collection of finite commutative rings satisfying a technical condition which limits the amount of torsion. There exists $\gamma \in (0,1)$ such that, for every $R \in \mathcal R$, every subset $A \subset R$ of cardinality at least $|R|^{1-\gamma}$ contains a nontrivial configuration $\{x,x+P_1(y_1,\ldots,y_n),\ldots, x+P_m(y_1,\ldots, y_n)\}$ for some $x,y_1,\ldots, y_n \in R$. The move from finite fields to finite commutative rings introduces many unexpected obstacles to our ergodic arguments, which I will give a flavor of.

4. 19 Omri Sarig

Title: Effective intrinsic ergodicity for surface diffeomorphisms

Abstract: A topologically transitive C infinity surface diffeomorphism with positive topological entropy has exactly one measure of maximal entropy (Newhouse; Buzzi-Crovisier-S.).

I will explain an "effective" version of this result: Any measure with entropy bigger than the topological entropy minus epsilon, is within distance O(\sqrt{\epsilon}) from the measure of maximal entropy. The distance is measured by comparing the integrals of smooth test functions. The square root is optimal.

This is joint work with Jerome Buzzi, Sylvain Crovisier, and Rene Ruhr.

5. 10 Jiyoung Han

Title: On the moment formulas for the primitive Siegel transform

Abstract: When we apply homogeneous dynamics to problems in the geometry of numbers, we first need to convert characteristic functions of Borel sets in vector spaces to functions on homogeneous spaces, and the Siegel transform is one of the classical tools for this. In this talk, I would like to introduce the definition of the primitive Siegel transform, which is related to counting primitive lattice points in the given bounded Borel set, and how to get its moment formulas 1) from those of the regular Siegel transform for the dimension being at least 3, and 2) by following the Schmidts’ work when the dimension is 2 (1960).

5. 17 Florian Richter

Title: On Erdos's sumset conjectures

Abstract: In the 1970's Erdos asked several questions about what kind of infinite arithmetic structures can be found in every set of natural numbers with positive density. In recent joint work with Bryna Kra, Joel Moreira, and Donald Robertson we use ergodic methods to resolve some of these long-standing problems. This talk will provide an overview of our results and describe some of the dynamical structures that are used to prove them.