Journal of Modern Dynamics (JMD)

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

Pages: 487 - 515, Issue 3, July 2010      doi:10.3934/jmd.2010.4.487

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Anatole Katok - Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States (email)
Federico Rodriguez Hertz - IMERL-Facultad de IngenierĂ­a, Universidad de la RepĂșblica, ulio Herrera y Reissig 565, CC 30, 11300 Montevideo, Uruguay (email)

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.

Keywords:  measure rigidity, nonuniform hyperbolicity, entropy, Lyapunov metric, synchronizing time change.
Mathematics Subject Classification:  37C40, 37D25, 37C85.

Received: January 2010;      Revised: August 2010;      Available Online: October 2010.