## Abstracts

**Jakub Byszewski**:

**Title:** *A density version of Cobham's theorem*

Abstract: Cobham’s theorem asserts that if a sequence is automatic with respect to two multiplicatively independent bases, then it is ultimately periodic. We prove a stronger density version of the result: if two sequences which are automatic with respect to two multiplicatively independent bases coincide on a set of density one, then they also coincide on a set of density one with a periodic sequence. We will discuss this result and its links with symbolic dynamics. This is joint work with Jakub Konieczny.

**Artem Dudko:**

**Title:** *On invariant random subgroups of approximately finite full groups*

**Abstract:** Approximately finite full groups are certain inductive limits of symmetric groups under block diagonal embeddings. An invariant random subgroup (IRS) of a group G is a Borel probability measure on the space of subgroups of G invariant under the action of G by conjugation. We classify ergodic IRS for a large class of simple approximately finite groups. The talk is based on a joint work with Konstantin Medynets.

**Aurelia Dymek:**

**Title:** *$\mathscr{B}$-free systems and dynamics*

**Abstract:** For a given subset $\mathscr{B}\subset\N$, we are interested in the associated set of $\mathscr{B}$-free numbers $\mathcal{F}_{\mathscr{B}}:=\Z\setminus\bigcup_{b\in\mathscr{B}}b\Z$. We consider the orbit closure of $\eta:=\raz_{\mathcal{F}_\mathscr{B}}\in\{0,1\}^\Z$ under the left shift $S$ and call it by a $\mathscr{B}$-free system. We will focus on topological dynamics of $\mathscr{B}$-free systems and recent results. We will formulate some open questions.

This talk will be based on the joint paper with Stanisław Kasjan, Joanna Kułaga-Przymus and Mariusz Lemańczyk.

**Yonatan Gutman:**

**Title:** *The embedding problem in topological dynamics*

Abstract: We survey old and new results related to the embedding problem in topological dynamics. More specifically we discuss conditions which guarantee that a $\mathbb{Z}$- dynamical system $(X,T)$, embeds into the $d$-cubical shift $(([0,1]^{d})^{\mathbb{Z}}, shift)$. Partially based on joint works with Lei Jin, Yixiao Qiao, Gábor Szabó and Masaki Tsukamoto.

**Victor Kleptsyn**:

**Title: ***Tools and theorems in one-dimensional dynamics*

**Abstract:** My course will be devoted to different tools and ideas in one-dimensional dynamics. Among the tools will see the distortion control technique, Lyapunov exponents, stationary measures and introduction of a random dynamics. We will see how the distortion control works in the classical Denjoy theorem and for the description of a parabolic implosion.

We will discuss the random dynamics on the circle and an important idea of a random contraction (Furstenberg-Antonov-Nalski-Deroin-VK-Navas).

We will see how stationary measure and Lyapunov exponents work in these cases.

If the time permits, we will discuss the random dynamics on the real line (Deroin, VK, Navas, Parwani), the intermediate regularity Denjoy theorem for the action of Z^d (Deroin, VK, Navas), or local vector fields argument by Loray-Rebelo-Nakai-Scherbakov.

**Dominik Kwietniak:**

**Title:** *All royal measures are loosely Kronecker*

**Abstract:** Gorodetski, Ilyashenko, Kleptsyn, and Nalsky constructed an ergodic invariant measure of a skew product torus map with zero Lyapunov exponent. Inspired by this construction, Bonatti, Diaz and Gorodetski gave sufficient conditions for weak convergence of a sequence of measures supported on periodic orbits to an ergodic measure. This method was later adapted by numerous authors and yielded examples of invariant measures with zero Lyapunov exponent in various settings. We call any measure obtained through this approach a royal measure. All royal measures are ergodic, but it was not known whether they share any other properties. In particular, it was an open question whether these measures can have positive entropy.

We show that royal measures always have zero entropy. Furthermore, we prove that all royal measures are loosely Kronecker. In other words, every royal measure is Kakutani equivalent (Kakutani equivalence is broader than the usual isomorphism of measure preserving systems) to an ergodic rotation of a non-discrete compact topological group. For the proof we introduce and study a new tool: the Feldman-Katok pseudometric fk-bar.

**Bill Mance:**

**Title:** *Normal numbers for the Cantor series expansion and possible applications in algebraic geometry*

**Abstract: **Normal numbers for the Cantor series expansion and possible applications in algebraic geometry Abstract: We will discuss basic properties of normal numbers for the Cantor series expansions and a recent result of D. Airey and B. Mance. Using ideas introduced in this paper and tools from descriptive set theory, it may be possible to show that information about algebraic varieties is encoded in the structure of sets of normal numbers. We will outline this idea and the barriers one may encounter in finishing it.