Journal of Modern Dynamics (JMD)

Entropic stability beyond partial hyperbolicity

Pages: 527 - 552, Issue 4, December 2013      doi:10.3934/jmd.2013.7.527

       Abstract        References        Full Text (505.3K)       Related Articles       

Jérôme Buzzi - C.N.R.S. & Département de Mathématiques, Université Paris-Sud, 91405 Orsay, France (email)
Todd Fisher - Department of Mathematics, Brigham Young University, Provo, UT 84602, United States (email)

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.

Keywords:  Measures of maximal entropy, topological entropy, robust ergodicity, ergodic theory, dominated splitting.
Mathematics Subject Classification:  37C40, 37A35, 37C15.

Received: May 2012;      Revised: August 2013;      Available Online: March 2014.