Abstract. We consider the set of maps f ∈ F α+ = ∪ β>α C 1+β of the circle which are covering maps of degree D, expanding, min x∈S 1 f (x) > 1 and orientation preserving. We are interested in characterizing the set of such maps f which admit a unique f -invariant probability measure µ minimizing ln f dµ over all f -invariant probability measures. We show there exists a set G + ⊂ F α+ , open and dense in the C 1+α -topology, admitting a unique minimizing measure supported on a periodic orbit. We also show that, if f admits a minimizing measure not supported on a finite set of periodic points, then f is a limit in the C 1+α -topology of maps admitting a unique minimizing measure supported on a strictly ergodic set of positive topological entropy.We use in an essential way a sub-cohomological equation to produce the perturbation. In the context of Lagrangian systems, the analogous equation was introduced by R. Mañé and A. Fathi extended it to the all configuration space in [8].We will also present some results on the set of f -invariant measures µ maximizing A dµ for a fixed C 1 -expanding map f and a general potential A, not necessarily equal to −ln f .

We generalize several results of the classical theory of Thermodynamic Formalism by considering a compact metric space M as the state space. We analyze the shift acting on M N and consider a general a-priori probability for defining the Transfer (Ruelle) operator. We study potentials A which can depend on the infinite set of coordinates in M N . We define entropy and by its very nature it is always a nonpositive number. The concepts of entropy and transfer operator are linked. If M is not a finite set there exist Gibbs states with arbitrary negative value of entropy. Invariant probabilities with support in a fixed point will have entropy equal to minus infinity. In the case M = S 1 , and the a-priori measure is Lebesgue dx, the infinite product of dx on (S 1 ) N will have zero entropy. We analyze the Pressure problem for a Hölder potential A and its relation with eigenfunctions and eigenprobabilities of the Ruelle operator. Among other things we analyze the case where temperature goes to zero and we show some selection results. Our general setting can be adapted in order to analyze the Thermodynamic Formalism for the Bernoulli space with countable infinite symbols. Moreover, the so called XY model also fits under our setting. In this last case M is the unitary circle S 1 . We explore the differentiable structure of (S 1 ) N by considering a certain class of smooth potentials and we show some properties of the corresponding main eigenfunctions.

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Received ? Revised ?Consider a α-Hölder function A : Σ → R and assume that it admits a unique maximizing measure µmax. For each β, we denote µ β , the unique equilibrium measure associated to βA. We show that (µ β ) satisfies a Large Deviation Principle, that is, for any cylinder C of Σ,where V (x) is any strict subaction of A.

We consider (M, d) a connected and compact manifold and we denote by Bi the Bernoulli space M Z . The analogous problem on the half-line N is also considered. Let A : Bi → R be an observable. Given a temperature T , we analyze the main properties of the Gibbs stateμ 1 T A . In order to do our analysis we consider the Ruelle operator associated to