# Abstracts

### Joel Moreira

Title: Multiple recurrence for polynomial and non-polynomial sequences

Abstract: The multiple recurrence theorem of Furstenberg proved in 1977 opened the doors to many subsequent extensions establishing recurrence along a variety of sequences. The study of more general multiple recurrence results led to remarkable developments in our understanding of the structure of measure preserving systems, and has interesting applications to number theory and combinatorics. I will present a new multiple recurrence theorem, recently obtained in joint work with Bergelson and Richter, which combines and unifies several previously known results, and implies new ones.

### Song Shao

Title: Topological characteristic factors and nilsystems

Abstract: In the study of multiple ergodic averages, the idea of characteristic factors plays a very important role. A general strategy for showing the existence of multiple ergodic average, is to find a factor such that the limiting behavior is unchanged when each function is replaced by its conditional expectation on this factor. Such a factor is called a characteristic factor. In this talk we will discuss the topological characteristic factors. We show that module an almost 1-1 extension, the maximal pro-nilfactor of a minimal dynamical system is the topological characteristic factor. Using results derived from the above fact, we are able to answer several open questions. For example, we show that the odd topological multiple recurrence theorem holds, and for a totally minimal system, there is a point whose orbit along the squares is dense. This is joint work with Eli Glasner, Wen Huang, Benjamin Weiss, and Xiangdong Ye.

### Nishant Chandgotia

Title: Predictive sets

Abstract: A subset of the integers $$P$$ is called predictive if for all zero-entropy processes $$X_i, \ i \in \mathbb{Z}$$, $$X_0$$ can be determined by $$X_i, \ i \in P$$. The classical formula for entropy shows that the set of natural numbers forms a predictive set. In joint work with Benjamin Weiss, we will explore some necessary and some sufficient conditions for a set to be predictive. These sets are related to Riesz sets (as defined by Y. Meyer) which arise in the study of singular measures. This and several questions will be discussed during the talk.

### Kostya Medynets

Title: Group Characters via Ergodic Actions

Abstract: A character of a finite group $$G$$ is defined as the trace of a matrix representation of the group $$G$$. This notion can be generalized to the class of infinite groups by looking at the properties of the matrix trace as a function. Namely, a function $$f: G \to \mathbb{C}$$ is a “normalized” character if $$f(1)=1, \ f(ab) = f(ba)$$ and $$f$$ is positive-definite. Vershik observed that by considering a measure preserving action of the group $$G$$ on a probability measure space $$(X,\mu)$$, the function $$f(g) = \mu(\text{FixedPoints}(g))$$ defines a character on the group $$G$$. In this talk we discuss several classes of groups for which the only characters are those that come from measure-preserving actions.

### Tamara Kucherenko

Title: Multiple phase transitions on compact symbolic systems

Abstract: A first-order phase transition refers to a loss of differentiability of the pressure function with respect to a parameter regarded as the inverse temperature. Such non-differentiability necessarily implies coexistence of several equilibrium states, although the converse is not true. In the case of Hölder continuous potentials on transitive SFTs the pressure is real analytic, and there are no phase transitions. Therefore, in order to allow the possibility of phase transitions one needs to consider potentials that are merely continuous. We present a method to explicitly construct a continuous potential on a full shift with any finite number of first order phase transitions occurring at any sequence of predetermined points. We are able to go even further. The convexity of the pressure implies that a continuous potential has at most countably many phase transitions. We show that the case of infinitely many phase transitions can indeed be realized. This is based on joint work with Anthony Quas and Christian Wolf.

### Francesco Cellarosi

Title: Central Limit Theorem for odometers and $$B$$-free integers

Abstract: Odometers (or von Neumann-Kakutani adding machines) are classical examples of dynamical systems of low complexity, much alike irrational rotations of the circle. We consider generalized adding machines. In spite of their rigid behaviour (zero entropy, not weakly mixing), we are able to prove a temporal Central Limit Theorem for the ergodic sums corresponding to certain (randomly chosen) observables, generalizing the work of M.B. Levin and E. Merzbach. As a consequence, we obtain a limit theorem for the dynamical systems naturally arising when studying the statistical properties of $$B$$-free integers. Joint work with Maria Avdeeva.