Measure rigidity for random dynamics on surfaces and related skew products

2015/06/22 by Brown, Aaron W., Hertz, Federico Rodriguez · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1506.06826

Abstract

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. 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-σ-algebra \mathcal F and derive a related result. A number of applications of our main theorem are presented.

Cited by

• Measure Rigidity beyond Homogeneous Dynamics

Related

• Measure rigidity for random dynamics on surfaces with positive entropy
• Random dynamics on real and complex projective surfaces
• Open sets of partially hyperbolic skew products having a unique SRB measure
• Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces
• Absolute continuity of stationary measures for random surface dynamics