Measure rigidity for random dynamics on surfaces and related skew products

Given a surface M and a Borel probability measure v on the group of C²-diffeomorphisms of M we study v-stationary probability measures on M. We prove for hyperbolic stationary measures the following trichotomy: 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 transformation equipped with a decreasing sub- σ -algebra F and derive a related result. A number of applications of our main theorem are presented.