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
MathSciNet review:
3671937
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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
MR Author ID:
912945
Email:
awb@uchicago.edu
Federico Rodriguez Hertz
Affiliation:
Department of Mathematics, The Pennsylvania State University, State College, Pennsylvania 16802
Email:
hertz@math.psu.edu
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