Absolute continuity of stationary measures

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].