For any countable group, and also for any locally compact second countable, compactly generated topological group, $G$, there exists a "universal" hypercyclic representation on a Hilbert space, in the sense that it simultaneously models every possible ergodic probability measure preserving free action of $G$. I will discuss the original proof of this theorem (a joint work with Benjy Weiss) and then, at the end of the talk, say some words about the development of this idea and its applications as expounded in a subsequent work of Sophie Grivaux.[-]

In this talk, we will first review some of the analogies between full groups of measure-preserving equivalence relations and the symmetric group over the integers, which have been used by A. Eisenmann and Y. Glasner to provide interesting examples of invariant random subgroups (IRSs) of the free group.
We will then see how the notion of cost, introduced by G. Levitt, naturally enters this picture. After that, we will explain how a stronger analogy between full groups and the symmetric group over the integers holds in the type III case.
A joint result with A. Kaïchouh which uses this analogy will be presented : full groups of hyperfinite type III equivalence relations have ample generics. This provides a positive answer to a question of A. Kechris and C. Rosendal on the existence of connected Polish group with ample generics.[-]

Let us say that a continuous linear operator $T$ acting on some Polish topological vector space is ergodic if it admits an ergodic probability measure with full support. This talk will be centred in the following question: how can we see that an operator is or is not ergodic? More precisely, I will try (if I'm able to manage my time) to talk about two “positive" results and one “negative" result. The first positive result says that if the operator $T$ acts on a reflexive Banach space and satisfies a strong form of frequent hypercyclicity, then $T$ is ergodic. The second positive result is the well-known criterion for ergodicity relying on the perfect spanning property for unimodular eigenvectors, of which I will outline a “soft" Baire category proof. The negative result will be stated in terms of a parameter measuring the maximal frequency with which (generically) the orbit of a hypercyclic vector for $T$ can visit a ball centred at 0. The talk is based on joint work with Sophie Grivaux.[-]

Subshifts on finite alphabets form a class of dynamical systems that bridge topological/ergodic dynamical systems with that of word combinatorics. In 1984, M. Boshernitzan used word combinatorics to provide a bound on the number of ergodic measures for a minimal subshift with bounds on its linear factor complexity growth rate. He further asked if the correct bound for subshifts naturally coded by interval exchange transformations (IETs) could be obtained by word combinatoric methods. (The ”correct” bound is roughly half that obtained by Boshernitzan's work.) In 2017 and joint with M. Damron, we slightly improved Boshernitzan's bound by restricting to a smaller class of subshifts that still contained IET subshifts. In recent work, we have further proved the ”correct” bound to subshifts whose languages satisfy a specific word combinatoric condition, which we called the Regular Bispecial Condition. (This condition is equivalent to being Eventually Dendric as independently introduced by F. Dolce and D. Perrin.)
During the same time we worked on our 2017 paper, V. Cyr and B. Kra were independently improving Boshernitzan's results. In 2019, they relaxed the conditions to no longer require minimality and extended Boshernitzan's bound to generic measures. (Generic measures are those that have generic points, meaning they satisfy the averaging limits as stated in Pointwise Ergodic Theorem. However, there are non-ergodic generic measures.) We have obtained the improved 2017 bound but for generic measures (and on a more general class of subshifts). It should be noted that, to our current knowledge, there does not exist a proof of the correct bound of generic measures for minimal IETs (by any method).In this talk, I will discuss these recent results and highlight related open problems.[-]