Measure rigidity for random dynamics on surfaces with positive entropy

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 $M$. Assuming the positivity of a certain entropy, the following dichotomy is proved: either the stable distributions for the random dynamics is non-random, or the measure is SRB. In the case that $\nu$-a.e. diffeomorphism preserves a common smooth measure $m$, we show that for any positive-entropy stationary measure $\mu$, either there exists a $\nu$-almost surely invariant $\mu$-measurable line field (corresponding do the stable distributions for almost every random composition) or the measure $\mu$ is $\nu$-almost surely invariant and coincides with an ergodic component of $m$. To prove the above result, we introduce a skew product with surface fibers over a measure preserving transformation equipped with an increasing sub-$\sigma$-algebra $\hat F$. Given an invariant measure $\mu$ for the skew product, and assuming the $\hat F$-measurability of the `past dynamics' and the fiber-wise conditional measures, we prove a dichotomy: either the fiber-wise stable distributions are measurable with respect to a related increasing sub-$\sigma$-algebra, or the measure $\mu$ is fiber-wise SRB.