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

Anatole Katok | Zendy; Federico Rodriguez Hertz | Zendy

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

Author(s) -

Anatole Katok,

Federico Rodriguez Hertz

Publication year - 2010

Publication title -

journal of modern dynamics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.668

H-Index - 25

eISSN - 1930-532X

pISSN - 1930-5311

DOI - 10.3934/jmd.2010.4.487

Subject(s) - mathematics , abelian group , lyapunov exponent , automorphism , pure mathematics , rigidity (electromagnetism) , invariant (physics) , torus , absolute continuity , invariant measure , entropy (arrow of time) , rank (graph theory) , mathematical analysis , combinatorics , ergodic theory , nonlinear system , mathematical physics , geometry , physics , structural engineering , quantum mechanics , engineering

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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