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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Measure rigidity for random dynamics on surfaces and related skew products
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by Aaron Brown and Federico Rodriguez Hertz PDF
J. Amer. Math. Soc. 30 (2017), 1055-1132 Request permission


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:
  • Federico Rodriguez Hertz
  • Affiliation: Department of Mathematics, The Pennsylvania State University, State College, Pennsylvania 16802
  • Email:
  • 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.
  • © Copyright 2017 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 30 (2017), 1055-1132
  • MSC (2010): Primary 37C40, 37H99; Secondary 37E30, 37D25, 28D15
  • DOI:
  • MathSciNet review: 3671937