2014/06/27 by Brown, Aaron W., Hertz, Federico Rodriguez
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1406.7201
Given a surface M and a Borel probability measure ν on the group of C2-diffeomorphisms of M, we study ν-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 ν-a.e. diffeomorphism preserves a common smooth measure m, we show that for any positive-entropy stationary measure μ, either there exists a ν-almost surely invariant μ-measurable line field (corresponding do the stable distributions for almost every random composition) or the measure μ is ν-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-σ-algebra F. Given an invariant measure μ for the skew product, and assuming the 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-σ-algebra, or the measure μ is fiber-wise SRB.