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

Anatole Katok; Federico Rodriguez Hertz

doi:10.3934/jmd.2010.4.487

Article Contents

2010, Volume 4, Issue 3: 487-515. Doi: 10.3934/jmd.2010.4.487

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

Anatole Katok^1, and
Federico Rodriguez Hertz^2,

1.
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
2.
IMERL-Facultad de Ingeniería, Universidad de la República, ulio Herrera y Reissig 565, CC 30, 11300 Montevideo

Received: December 31, 2009

Revised: July 31, 2010

Published: September 2010

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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.

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Mathematics Subject Classification: 37C40, 37D25, 37C85.

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References

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