Measure rigidity for random dynamics on surfaces and related skew products

TL;DR

This paper investigates the behavior of stationary measures for random surface diffeomorphisms, establishing a trichotomy for hyperbolic measures and exploring skew products with surface fibers, with various applications.

Contribution

It introduces a trichotomy for hyperbolic stationary measures and analyzes skew products with surface fibers, advancing understanding of measure rigidity in random surface dynamics.

Findings

Hyperbolic stationary measures exhibit a three-way classification.

Stable distributions are either non-random, SRB, or supported on finite sets.

Applications demonstrate the theorem's relevance to surface dynamics.

Abstract

Given a surface $M$ and a Borel probability measure $\nu$ on the group of $C^2$-diffeomorphisms of $M$, we study $\nu$-stationary probability measures on $M$. We prove for hyperbolic stationary measures the following trichotomy: either the stable distributions are non-random, the measure is SRB, or the measure is supported on a finite set and is hence almost-surely invariant. In the proof of the above results, we study skew products with surface fibers over a measure preserving transformations equipped with a decreasing sub-$\sigma$-algebra $\hat {\mathcal F}$ and derive a related result. A number of applications of our main theorem are presented.