Expanding on average random dynamics

Date: Tuesday 14th October, 2025

Abstract

We consider exponential mixing for volume preserving random dynamical systems on surfaces. Suppose that $(f_1, ..., f_m)$ is a tuple of volume preserving diffeomorphisms of a closed surface $M$. We now consider the uniform Bernoulli random dynamical system that this tuple generates on $M$. We assume that this tuple satisfies a condition called being ``expanding on average," which means that there exist $C > 0$ and a natural number $N$ such that for all unit tangent vectors $v$, $\mathbb{E}[\ln  || Df^N v|| ] > C$ , where the expectation is taken over all the realizations of the random dynamics. From this assumption we show quenched exponential mixing. (This is joint work with Dmitry Dolgopyat)