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

MEASURE AND COCYCLE RIGIDITY FOR CERTAIN NONUNIFORMLY HYPERBOLIC ACTIONS OF HIGHER-RANK ABELIAN GROUPS

Anatole Katok
, Federico Rodriguez Hertz

Research output: Chapter in Book/Report/Conference proceeding › Chapter

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 language	English (US)
Title of host publication	The Collected Works of Anatole Katok
Subtitle of host publication	In 2 Volumes
Publisher	World Scientific Publishing Co.
Pages	2433-2461
Number of pages	29
Volume	2
ISBN (Electronic)	9789811238079
ISBN (Print)	9789811238062
DOIs	https://doi.org/10.3934/jmd.2010.4.487
State	Published - Jan 1 2024

All Science Journal Classification (ASJC) codes

General Mathematics
General Engineering

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10.3934/jmd.2010.4.487

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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.",

author = "Anatole Katok and Hertz, \{Federico Rodriguez\}",

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AB - 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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ER -

MEASURE AND COCYCLE RIGIDITY FOR CERTAIN NONUNIFORMLY HYPERBOLIC ACTIONS OF HIGHER-RANK ABELIAN GROUPS

Abstract

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