Renaud Leplaideur

A dynamical proof for the convergence of Gibbs measures at temperature zero

We give a dynamical proof of a result due to Bremont (2003 Nonlinearity 16 419–26). It concerns the problem of maximizing measures for some given observable : for a subshift of finite type, and when only depends on a finite number of coordinates, it was proved in Bremont (2003) that the unique equilibrium state associated with β converges to some measure when β goes to +∞. This measure has maximal entropy among the maximizing measures for . We give here a dynamical proof of this result and we improve it. We prove that for any Holder continuous function (not necessarily locally constant), f, the unique equilibrium state associated with f + β converges to some measure with maximal f-pressure among the maximizing measures. Moreover we also identify the limit measure.

Local product structure for Equilibrium States

The usual way to study the local structure of Equilibrium State of an AxiomA 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 Holder-continuous functions are related to other Equilibrium States of some special sub-systems satisfying a sort of expansiveness. Using different kinds of extensions the local product structure of Gibbs-measure is proven.

Selection of Ground States in the Zero Temperature Limit for a One-Parameter Family of Potentials

For the subshift of nite type = f0; 1; 2g N we study the convergence and the selection at temperature zero of the Gibbs measure associated to a non-locally constant Holder potential which admits exactly two maximizing ergodic measures. These measures are Dirac measures at two dierent xed points and the potential is atter at one of these two xed points. We prove that there always is convergence but not necessarily to the Dirac measure at the point where the potential is the attest. This is contrary to what was expected in the light of the analogous problem in Aubry-Mather theory (1). This is also contrary to the nite range case where the equilibrium state converges to the equi-barycentre of the two Dirac measures. Moreover we emphasize the strange behavior of the Gibbs measure: the eigenmeasure selects one Dirac measure ( at the point where the potential is the attest) and the eigen- function selects the other one (at the point where the potential is the sharpest).

Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes

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 with Holder continuous potentials.

Flatness Is a Criterion for Selection of Maximizing Measures

For the one-dimensional classical spin system, each spin being able to get Np+1 values, and for a non-positive potential, locally proportional to the distance to one of N disjoint configurations set {(j−1)p+1,…,jp}ℤ, 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 measures and whose supports are the two shifts where the potential is the flattest.In particular, this is a hint to solve the open problem of selection, and this indicates that flatness is probably a/the criterion for selection as it was conjectured by A.O. Lopes.As a by product we get convergence of the eigenfunction at the log-scale to a unique calibrated subaction.

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

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 −t log 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. AMSC : 37C29, 37C45, 37D25, 37D35.

CENTRAL LIMIT THEOREM FOR DIMENSION OF GIBBS MEASURES IN HYPERBOLIC DYNAMICS

For an equilibrium measure of a Holder 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. A noticeable consequence is that when this measure is not absolutely continuous, the probability that a ball of radius e chosen at random have a measure smaller (or larger) than eδ is asymptotically equal to 1/2, where δ is the Hausdorff dimension of the measure. Our method applies to a class of non-conformal expanding maps on the d-dimensional torus. It also applies to conformal repellers and Axiom A surface diffeomorphisms and possibly to a class of one-dimensional non-uniformly expanding maps. These generalizations are presented at the end of the paper.