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Equilibrium states for partially hyperbolic horseshoes

Published online by Cambridge University Press:  18 January 2010

R. LEPLAIDEUR ,
K. OLIVEIRA  and
I. RIOS
R. LEPLAIDEUR
Affiliation:
Laboratoire de Mathématiques, UMR 6205, Université de Bretagne Occidentale, 6 rue Victor le Gorgeu, BP 809, F-29285 Brest, Cedex, France (email: Renaud.Leplaideur@univ-brest.fr)
K. OLIVEIRA
Affiliation:
Departamento de Matemática, Universidade Federal de Alagoas, Maceió, 57072-090, Brazil (email: krerley@gmail.com)
I. RIOS
Affiliation:
Instituto de Mathemática, Universidade Federal Fluminense, Rua Mário Santos Braga s/n, Niterói, RJ, 24.020-140, Brazil (email: rios@impa.br)
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Abstract

We study ergodic properties of invariant measures for the partially hyperbolic horseshoes, introduced in Díaz et al [Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergod. Th. & Dynam. Sys. 29 (2009), 433–474]. These maps have a one-dimensional center direction Ec, and are at the boundary of the (uniformly) hyperbolic diffeomorphisms (they are constructed bifurcating hyperbolic horseshoes via heterodimensional cycles). We prove that every ergodic measure is hyperbolic, but the set of Lyapunov exponents in the central direction has gap: all ergodic invariant measures have negative exponent, with the exception of one ergodic measure with positive exponent. As a consequence, we obtain the existence of equilibrium states for any continuous potential. We also prove that there exists a phase transition for the smooth family of potentials given by ϕt=t log ∣DF∣Ec∣.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 1 , February 2011 , pp. 179 - 195
Copyright
Copyright © Cambridge University Press 2009

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