Ergodicity of (co)expanding on average random dynamical systems

TL;DR

This paper establishes ergodicity for certain random dynamical systems with expansion properties, demonstrating stable ergodicity on spheres generated by dense subgroups of rotation groups, with implications for spectral gaps and statistical theorems.

Contribution

It proves ergodicity under expansion and irreducibility conditions and extends known results to odd dimensions, including systems with zero Lyapunov exponents.

Findings

Ergodicity proven for random systems with expansion and irreducibility.

Stable ergodicity shown for rotations generating dense subgroups on spheres.

Spectral gap and statistical limit theorems established for these systems.

Abstract

We prove ergodicity for random dynamics satisfying some expansion and irreducibility conditions. As a particular application, we show that if $R_1,R_2\in \mathrm{SO}(d+1)$, $d\ge 2$, 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.