Absolute continuity of stationary 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 $\hat{I}¼$, supported on the union of these neighborhoods, and verifying certain conditions, the unique $\hat{I}¼$-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].

34 pages