Entropic stability beyond partial hyperbolicity

Jérôme Buzzi; Todd Fisher

doi:10.3934/jmd.2013.7.527

Article Contents

2013, Volume 7, Issue 4: 527-552. Doi: 10.3934/jmd.2013.7.527

This issue Previous Article Bowen's construction for the Teichmüller flow Next Article A generic-dimensional property of the invariant measures for circle diffeomorphisms

Entropic stability beyond partial hyperbolicity

Jérôme Buzzi^1, and
Todd Fisher^2,

1.
C.N.R.S. & Département de Mathématiques, Université Paris-Sud, 91405 Orsay
2.
Department of Mathematics, Brigham Young University, Provo, UT 84602

Received: April 30, 2012

Revised: July 31, 2013

Published: February 2014

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Abstract

We analyze a class of $C^0$-small but $C^1$-large deformations of Anosov diffeomorphisms that break the topological conjugacy and structural stability, but unexpectedly retain the following stability property. The usual semiconjugacy mapping the deformation to the Anosov diffeomorphism is in fact an isomorphism with respect to all ergodic, invariant probability measures with entropy close to the maximum. In particular, the value of the topological entropy and the existence of a unique measure of maximal entropy are preserved. We also establish expansiveness around those measures. However, this expansivity is too weak to ensure the existence of symbolic extensions.
Many constructions of robustly transitive diffeomorphisms can be done within this class. In particular, we show that it includes a class described by Bonatti and Viana of robustly transitive diffeomorphisms that are not partially hyperbolic.

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Mathematics Subject Classification: 37C40, 37A35, 37C15.

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References

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