Analysis and Probability seminar

Abstract: Let $G$ be a countable discrete group, and let $\mu$ be a probability measure on $G$ with finite (Shannon) entropy. We use ideas from harmonic analysis to initiate the study of several related concepts associate to a probability measure $\mu$ and exploit their relations. First, we look at the concept of Lyapunov exponent of $\mu$ with respect to weights on $G$ and build a framework that connects it to the entropy of $\mu$ in $G$. This is done by introducing a generalization of Avez entropy of $\mu$, taking into account the given weight, and investigating in details their relations together as well as to the actions of $G$ on measurable stationary spaces.

As a byproduct of our techniques, we show that for a large class of groups (e.g. groups with rapid decay) and probability measures on them, the weak containment of the representation $\pi_X$ of $G$ on a $\mu$-stationary space $(X,\xi)$ implies that the Furstenberg entropy of $\mu$ in $(X,\xi)$ conicides with the Avez entropy of $\mu$. Hence these types of stationary spaces are precisely measure-preserving extension of the Poisson boundary of $(G,\mu)$. In particular, $(X,\xi)$ is an amenable $(G,\mu)$-space.

This is a joint work with Benjamin Anderson-Sackaney, Tim de Laat, and Matthew Wiersma.