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Ergodic Theory and Dynamical Systems Ergodic Theory and Dynamical Systems

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On t-conformal measures and Hausdorff dimension for a family of non-uniformly hyperbolic horseshoes

Published online by Cambridge University Press:  02 March 2009

RENAUD LEPLAIDEUR  and
ISABEL RIOS
RENAUD 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)
ISABEL RIOS
Affiliation:
Instituto de Matemática, Universidade Federal Fluminense, Rua Mário Santos Braga s/n, Niterói, RJ, 24.020-140, Brasil (email: rios@mat.uff.br)
Corresponding
E-mail address:
Renaud.Leplaideur@univ-brest.fr
rios@mat.uff.br
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Abstract

In this paper we consider horseshoes with homoclinic tangencies inside the limit set. For a class of such maps, we prove the existence of a unique equilibrium state μt, associated to the (non-continuous) potential −tlog Ju. We also prove that the Hausdorff dimension of the limit set, in any open piece of unstable manifold, is the unique number t0 such that the pressure of μt0 is zero. To deal with the discontinuity of the jacobian, we introduce a countable Markov partition adapted to the dynamics, and work with the first return map defined in a rectangle of it.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 6 , December 2009 , pp. 1917 - 1950
Copyright
Copyright © Cambridge University Press 2009

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