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.[-]

The behaviour of infinite translation surfaces is, in many regards, very different from the finite case. For example, the geodesic flow is often not recurrent or is not even defined for infinite time in a generic direction.
However, we show that if one focuses on a class of infinite translation surfaces that exclude the obvious counter-examples, one can adapted the proof of Kerckhoff, Masur, and Smillie and show that the geodesic flow is uniquely ergodic in almost every direction. We call this class of surface essentially finite.
(joint work with Anja Randecker).[-]

We consider a Markov process living in some space E, and killed (penalized) at a rate depending on its position. In the last decade, several conditions have been given ensuring that the law of the process conditioned on survival converges to a quasi-stationary distribution exponentially fast in total variation distance. In this talk, we will present very simple examples of penalized Markov process whose conditional law cannot converge in total variation, and we will give a sufficient condition implying contraction and convergence of the conditional law in Wasserstein distance to a unique quasi-stationary distribution. Our criterion also imply a first-order expansion of the probability of survival, the ergodicity in Wasserstein distance of the Q-process, i.e. the process conditioned to never be killed, and quasi-ergodicity in Wasserstein distance. We then apply this criterion to several examples, including Bernoulli convolutions and piecewise deterministic Markov processes of the form of switched dynamical systems, for which convergence in total variation is not possible.
This is joint work with Edouard Strickler (CNRS, Université de Lorraine) and Denis Villemonais (Université de Lorraine).[-]

Concentration is an important property of independent random variable, showing that any reasonable function of such variables does not vary a lot around its mean. Observables generated by the iteration of a chaotic enough dynamical system often share a lot of properties with independent random variables. In this survey talk, we discuss several situations where one can prove concentration for them, in uniformly or non-uniformly hyperbolic situations. We also explain why such a property is important to answer relevant geometric or dynamical questions.
concentration - martingales - dynamical systems - Young towers - uniform hyperbolicity - moment bounds[-]

We study law of rare events for random dynamical systems. We obtain an exponential law (with respect to the invariant measure of the skew-product) for super-polynomially mixing random dynamical systems.
For random subshifts of finite type, we analyze the distribution of hitting times with respect to the sample measures. We prove that with a superpolynomial decay of correlations one can get an exponential law for almost every point and with stronger mixing assumptions one can get a law of rare events depending on the extremal index for every point. (These are joint works with Benoit Saussol and Paulo Varandas, and Mike Todd).[-]