Renormalization, Thermodynamic Formalism and Quasi-Crystals in Subshifts

We examine thermodynamic formalism for a class of renormalizable dynamical systems which in the symbolic space is generated by the Thue-Morse substitution, and in complex dynamics by the Feigenbaum-Coullet-Tresser map. The basic question answered is whether fixed points $V$ of a renormalization operator \CR acting on the space of potentials are such that the pressure function \gamma \mapsto \CP(-\gamma V) exhibits phase transitions. This extends the work by Baraviera, Leplaideur and Lopes on the Manneville-Pomeau map, where such phase transitions were indeed detected. In this paper, however, the attractor of renormalization is a Cantor set (rather than a single fixed point), which admits various classes of fixed points of \CR, some of which do and some of which do not exhibit phase transitions. In particular, we show it is possible to reach, as a ground state, a quasi-crystal before temperature zero by freezing a dynamical system.Comment: The paper was withdrawn from publication due to an error found in some proof. This is a new version and resubmitted for publication. The occurance of phase transition is proved for a parameter a<1 and it is proved there is no phase transition for a>1. For the value a=1 it is still unkow

Central limit theorem for dimension of Gibbs measures for skew expanding maps

We consider a class of non-conformal expanding maps on the $d$-dimensional torus. For an equilibrium measure of an H\"older 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 half of the balls of radius \eps have a measure smaller than \eps^\delta and half of them have a measure larger than \eps^\delta, where $\delta$ is the Hausdorff dimension of the measure. We first show that the problem is equivalent to the study of the fluctuations of some Birkhoff sums. Then we use general results from probability theory as the weak invariance principle and random change of time to get our main theorem. Our method 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

Large deviations for return times in non-rectangle sets for axiom A diffeomorphisms

For Axiom A diffeomorphisms and equilibrium states, we prove a Large deviations result for the sequence of successive return times into a fixed Borel set, under some assumption on the boundary. Our result relies on and extends the work by Chazottes and Leplaideur who considered cylinder sets of a Markov partition

Thermodynamic formalism for Lorenz maps

For a 2-dimensional map representing an expanding geometric Lorenz at- tractor we prove that the attractor is the closure of a union of as long as possible unstable leaves with ending points. This allows to define the notion of good measures, those giving full measure to the union of these open leaves. Then, for any H\"older continuous potential we prove that there exists at most one relative equilibrium state among the set of good measures. Condition yielding existence are given.Comment: 36 page

Large deviation for return times in open sets for axiom A diffeomorphisms

For axiom A diffeomorphisms and equilibrium state, we prove a Large deviation result for the sequence of successive return times into a fixed open set, under some assumption on the boundary. Our result relies on and extends the work by Chazottes and Leplaideur who where considering cylinder sets of a Markov partition

RENORMALIZATION, THERMODYNAMIC FORMALISM AND QUASI-CRYSTALS IN SUBSHIFTS.

The paper was withdrawn from publication due to an error found in some proof. This is a new version and resubmitted for publication. The occurance of phase transition is proved for a parameter a1. For the value a=1 it is still unkown.We examine thermodynamic formalism for a class of renormalizable dynamical systems which in the symbolic space is generated by the Thue-Morse substitution, and in complex dynamics by the Feigenbaum-Coullet-Tresser map. The basic question answered is whether fixed points $V$ of a renormalization operator \CR acting on the space of potentials are such that the pressure function \gamma \mapsto \CP(-\gamma V) exhibits phase transitions. This extends the work by Baraviera, Leplaideur and Lopes on the Manneville-Pomeau map, where such phase transitions were indeed detected. In this paper, however, the attractor of renormalization is a Cantor set (rather than a single fixed point), which admits various classes of fixed points of \CR, some of which do and some of which do not exhibit phase transitions. In particular, we show it is possible to reach, as a ground state, a quasi-crystal before temperature zero by freezing a dynamical system