Absolute continuity of stationary measures for random surface dynamics

TL;DR

This paper establishes conditions under which stationary measures for random surface dynamics are absolutely continuous, covering systems with dissipative diffeomorphisms and perturbations of volume-preserving maps, with applications to toral automorphisms.

Contribution

It introduces new criteria involving uniform expansion on average for absolute continuity of stationary measures in random surface dynamics.

Findings

Stationary measures are either atomic or absolutely continuous.

Results apply to systems generated by perturbations of toral automorphisms.

Orbit classification and equidistribution are achieved.

Abstract

We find conditions for stationary measures of random dynamical systems on surfaces having dissipative diffeomorphisms to be absolutely continuous. These conditions involve a uniformly expanding on average property in the future (UEF) and past (UEP). Our results can cover random dynamical systems generated by "very dissipative" diffeomorphisms and perturbations of volume preserving surface diffeomorphisms. For example, we can consider a random dynamical system on $\mathbb{T}^2$ generated by perturbations of a pair of non-commuting infinite order toral automorphisms with any arbitrary single diffeomorphism. In this case, we conclude that stationary measures are either atomic or absolutely continuous. We also obtain an orbit classification and equidistribution result.