Measure rigidity for random dynamics on surfaces and related skew products

By Aaron W. Brown, F. R. Hertz. 2015

49 citations recorded by the source corpus.

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 $M$. We prove for hyperbolic stationary measures the following trichotomy: either 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 transformations equipped with a decreasing sub-$\sigma$-algebra $\hat {\mathcal F}$ and derive a related result. A number of applications of our main theorem are presented.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/91cf0473284a011e3a10db41c55eacc4c984a642

Citations in the corpus

• Uniqueness (2021, unknown): “These results are inspired by some important measure rigidity techniques from [EM18, EL, BQ11, BRH17] to classify stationary measures of certain random products (see also [CD20] for an interesting application of [BRH17]).”