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Measure rigidity for random dynamics on surfaces and related skew products


Authors: Aaron Brown and Federico Rodriguez Hertz
Journal: J. Amer. Math. Soc. 30 (2017), 1055-1132
MSC (2010): Primary 37C40, 37H99; Secondary 37E30, 37D25, 28D15
DOI: https://doi.org/10.1090/jams/877
Published electronically: March 7, 2017
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Abstract: 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.


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Additional Information

Aaron Brown
Affiliation: Department of Mathematics, The University of Chicago, Chicago, Illinois 60637
Email: awb@uchicago.edu

Federico Rodriguez Hertz
Affiliation: Department of Mathematics, The Pennsylvania State University, State College, Pennsylvania 16802
Email: hertz@math.psu.edu

DOI: https://doi.org/10.1090/jams/877
Keywords: Measure rigidity, non-uniform hyperbolicity, stiffness of stationary measures, random dynamics, SRB measures
Received by editor(s): July 31, 2015
Received by editor(s) in revised form: October 10, 2016
Published electronically: March 7, 2017
Additional Notes: The first author was supported by an NSF postdoctoral research fellowship DMS-1104013.
The second author was supported by NSF grants DMS-1201326 and DMS-1500947.
Article copyright: © Copyright 2017 American Mathematical Society