Ergodicity of (co)expanding on average random dynamical systems

Jonathan DeWitt; Dmitry Dolgopyat; Zhiyuan Zhang

paper

Ergodicity of (co)expanding on average random dynamical systems

arXiv:2605.21199

Abstract

We prove ergodicity for random dynamics satisfying some expansion and irreducibility conditions. As a particular application, we show that if $R_{1}, R_{2} \in SO (d + 1)$ , $d \geq 2$ , generate a dense subgroup, then the random dynamics of $R_{1}$ and $R_{2}$ on $S^{d}$ is stably ergodic. Previously this was only known to hold in even dimensions. As a consequence, we deduce spectral gap and statistical limit theorems for such systems. In particular, our results apply in the presence of zero Lyapunov exponents.