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

Renaud Leplaideur (1, 2) , Isabel Rios

1 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique
2 ISEA - Institut de sciences exactes et appliquées

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 μ t , associated to the (non-continuous) potential − t log J u . We also prove that the Hausdorff dimension of the limit set, in any open piece of unstable manifold, is the unique number t 0 such that the pressure of μ t 0 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.

Domaines: Systèmes dynamiques [math.DS]

Renaud Leplaideur, Isabel Rios. On t -conformal measures and Hausdorff dimension for a family of non-uniformly hyperbolic horseshoes. Ergodic Theory and Dynamical Systems, 2009, 29 (6), pp.1917-1950. ⟨10.1017/S0143385708000941⟩. ⟨hal-03462645⟩