Measure rigidity for random dynamics on surfaces and related skew products

Aaron W Brown, Federico Rodriguez Hertz

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


Given a surface M and a Borel probability measure v on the group of C2-diffeomorphisms of M we study v-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- σ -algebra F and derive a related result. A number of applications of our main theorem are presented.

Original languageEnglish (US)
Pages (from-to)1055-1132
Number of pages78
JournalJournal of the American Mathematical Society
Issue number4
StatePublished - 2017


  • Measure rigidity
  • Non-uniform hyperbolicity
  • Random dynamics
  • SRB measures
  • Stiffness of stationary measures

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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