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Thermodynamic formalism for a family of non-uniformly hyperbolic horseshoes and the unstable Jacobian

Published online by Cambridge University Press:  17 March 2010

RENAUD LEPLAIDEUR
RENAUD LEPLAIDEUR*
Affiliation:
Laboratoire de Mathématiques, UMR 6205, Université de Bretagne Occidentale, 6 av. Victor Le Gorgeu, CS 93837, F-29238 BREST Cedex 3, France (email: renaud.leplaideur@univ-brest.fr)
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Abstract

In this article we prove the existence and uniqueness of equilibrium states for the potential and the class of non-uniformly hyperbolic horseshoes which was introduced in Rios [Unfolding homoclinic tangencies inside horseshoes: hyperbolicity, fractal dimensions and persistent tangencies. Nonlinearity14 (2001), 431–462]. We show that the pressure t↦𝒫(t) for −tlog Ju is real-analytic on . We give the exact equations of the two asymptotes to the graph of 𝒫(t) at ±∞ and we prove that these non-uniformly hyperbolic horseshoes do not have measures which minimize the unstable Lyapunov exponent.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 2 , April 2011 , pp. 423 - 447
Copyright
Copyright © Cambridge University Press 2010

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