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- R Leplaideur, K Oliveira, I Rios
- 2008

In this paper, we study ergodic features of invariant measures for the partially hyperbolic horseshoe at the boundary of uniformly hyperbolic diffeomorphisms constructed in [12]. Despite the fact that the non-wandering set is a horseshoe, it contains intervals. We prove that every recurrent point has non-zero Lyapunov exponents and all ergodic invariant… (More)

The usual way to study the local structure of Equilibrium State of an Axiom-A diffeomorphism or flow is to use the symbolic dynamic and to push results on the manifold. A new geometrical method is given. It consists in proving that Equilibrium States for Hölder-continuous functions are related to other Equilibrium States of some special subsystems… (More)

In this paper we consider horseshoes containing an orbit of homoclinic tangency accumulated by periodic points. We prove a version of the Invariant Manifolds Theorem, construct finite Markov partitions and use them to prove the existence and uniqueness of equilibrium states associated to Hölder continuous potentials.

We exhibit examples of mixing subshifts of finite type and potentials such that there are phase transitions but the pressure is always strictly convex. More surprisingly , we show that the pressure can be analytic on some interval although there exist several equilibrium states.

We present a method to construct equilibrium states via induction. This method can be used for some non-uniformly hyperbolic dynamical systems and for non-Hölder continuous potentials. It allows to prove the occurrence of phase transition. 1.1. Goal. We consider a dynamical system (X, f), where X is a compact metric space and f is topology mixing and local… (More)

We examine the renormalization operator determined by the Fi-bonacci substitution. We exhibit a fixed point and determine its stable leaf (under iteration of the operator). Then, we study the thermodynamic formalism for potentials in this stable leaf, and prove they have a freezing phase transition, with ground state supported on the attracting… (More)

For the subshift of finite type Σ = {0, 1, 2} N we study the convergence and the selection at temperature zero of the Gibbs measure associated to a non-locally constant Hölder potential which admits exactly two maximizing ergodic measures. These measures are Dirac measures at two different fixed points and the potential is flatter at one of these two fixed… (More)

We consider a diffeomorphism f of a compact manifold M which is Almost Axiom A, i.e. f is hyperbolic in a neighborhood of some compact f-invariant set, except in some singular set of neutral points. We prove that if there exists some f-invariant set of hyperbolic points with positive unstable-Lebesgue measure such that for every point in this set the stable… (More)

For a full shift with N p + 1 symbols and for a non-positive potential, locally proportional to the distance to one of N disjoint full shifts with p symbols, we prove that the equilibrium state converges as the temperature goes to 0. The main result is that the limit is a convex combination of the two ergodic measures with maximal entropy among maximizing… (More)

- Renaud Leplaideur, Benoˆıt Saussol
- 2009

We consider a class of non-conformal expanding maps on the d-dimensional torus. For an equilibrium measure of an Hölder potential, we prove an analogue of the Central Limit Theorem for the fluctuations of the logarithm of the measure of balls as the radius goes to zero. An unexpected consequence is that when the measure is not absolutely continuous , then… (More)