Absolute continuity of stationary measures

TL;DR

This paper establishes conditions under which stationary measures for certain volume-preserving Anosov diffeomorphisms are absolutely continuous, extending understanding of measure regularity in hyperbolic dynamical systems.

Contribution

It provides new criteria for absolute continuity of stationary measures near Anosov diffeomorphisms sharing stable and unstable cones.

Findings

Existence of neighborhoods where stationary measures are absolutely continuous.

Conditions on measures supported in these neighborhoods.

Application of equidistribution results to these measures.

Abstract

Let $f$ and $g$ be two volume preserving, Anosov diffeomorphisms on $\mathbb{T}^2$, sharing common stable and unstable cones. In this paper, we find conditions for the existence of (dissipative) neighborhoods of $f$ and $g$, $\mathcal{U}_f$ and $\mathcal{U}_g$, with the following property: for any probability measure $\mu$, supported on the union of these neighborhoods, and verifying certain conditions, the unique $\mu$-stationary SRB measure is absolutely continuous with respect to the ambient Haar measure. Our proof is inspired in the work of Tsujii for partially hyperbolic endomorphisms [Tsu05]. We also obtain some equidistribution results using the main result of [BRH17].