Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups

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Abstract

We prove absolute continuity of high-entropy hyperbolic invariant measures for smooth actions of higher-rank abelian groups assuming that there are no proportional Lyapunov exponents. For actions on tori and infranilmanifolds the existence of an absolutely continuous invariant measure of this kind is obtained for actions whose elements are homotopic to those of an action by hyperbolic automorphisms with no multiple or proportional Lyapunov exponents. In the latter case a form of rigidity is proved for certain natural classes of cocycles over the action.

Original languageEnglish (US)
Pages (from-to)487-515
Number of pages29
JournalJournal of Modern Dynamics
Volume4
Issue number3
DOIs
StatePublished - Jul 1 2010

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

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