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Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems

Published online by Cambridge University Press:  10 June 2011

J. BUZZI ,
T. FISHER ,
M. SAMBARINO  and
C. VÁSQUEZ
J. BUZZI
Affiliation:
C.N.R.S. and Département de Mathématiques, Université Paris-Sud, 91405 Orsay, France (email: jerome.buzzi@math.u-psud.fr)
T. FISHER
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, USA (email: tfisher@math.byu.edu)
M. SAMBARINO
Affiliation:
CMAT-Facultad de Ciencias, U. de la Republica, Montevideo, Uruguay (email: samba@cmat.edu.uy)
C. VÁSQUEZ
Affiliation:
Instituto de Matemática, Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile (email: carlos.vasquez@ucv.cl)
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Abstract

We show that a class of robustly transitive diffeomorphisms originally described by Mañé are intrinsically ergodic. More precisely, we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic and structurally stable, but nevertheless have the following stability with respect to their entropy. Their topological entropy is constant and they each have a unique measure of maximal entropy with respect to which periodic orbits are equidistributed. Moreover, equipped with their respective measure of maximal entropy, these diffeomorphisms are pairwise isomorphic. We show that the method applies to several classes of systems which are similarly derived from Anosov, i.e. produced by an isotopy from an Anosov system, namely, a mixed Mañé example and one obtained through a Hopf bifurcation.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 1 , February 2012 , pp. 63 - 79
Copyright
Copyright © Cambridge University Press 2011

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