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.[-]

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.[-]