Measure rigidity for random dynamics on surfaces and related skew products

Brown, Aaron¹ ; Hertz, Federico²

¹ Department of Mathematics, The University of Chicago, Chicago, Illinois 60637
² Department of Mathematics, The Pennsylvania State University, State College, Pennsylvania 16802

Journal of the American Mathematical Society, Tome 30 (2017) no. 4, pp. 1055-1132

Résumé

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: 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-$\sigma$-algebra $\hat {\mathcal F}$ and derive a related result. A number of applications of our main theorem are presented.

Détail

DOI : 10.1090/jams/877

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