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

Abstract. In an explicit family of partially hyperbolic diffeomorphisms of the torus T 3 , Shub and Wilkinson recently succeeded in perturbing the Lyapunov exponents of the center direction. We present here a local version of their argument, allowing one to perturb the center Lyapunov exponents of any partially hyperbolic system, in any dimension and with arbitrary dimension of the center bundle. IntroductionOne of the classical problems of the theory of dynamical systems is the understanding of the Lyapunov exponents. From Oseledets' Theorem we know that Lyapunov exponents are defined for almost every point with respect to any given invariant measure, and are independent of the point if moreover the measure is ergodic. Pesin's theory recovers some hyperbolic behavior for the points whose Lyapunov exponents are all non-zero (see for instance [FHY]). In particular, these points have well-defined unstable and stable invariant manifolds. For these reasons, an ergodic invariant measure µ is called hyperbolic if all its exponents are different from zero.On the other hand, the presence of zero exponents creates many obstacles to a good ergodic description of the system and is often related to some pathologies. For this reason, it is important to understand in which situations the zero exponents could be removed by perturbations.In the negative direction a recent result by Bochi [Boc] shows that, for C 1 -generic nonAnosov conservative diffeomorphisms on compact surfaces, almost every point has zero Lyapunov exponents. For a higher dimensional version see [BocVi]. We hope that this kind of result is typical only of the C 1 -topology and that zero Lyapunov exponents are no longer generic for more regular systems. It is possible to illustrate this contrast in the a priori simpler case of linear cocycles: from Bochi we know that zero exponents are generic for non-hyperbolic continuous cocycles of SL(2, R) but this is no longer true assuming that the cocycle satisfies a Hölder condition; see [BoVi], [BoGVi].

For the shift σ in Σ = {0, 1} N , we define the renormalization for potentials byWe show that for a good H, there is a unique fixed point for R. It is the Hofbauer potential V * .We show that the stable set of the Hofbauer potential, i.e. the set of potentials V such that R n (V ) converges to V * is characterized by the germ of these potentials close to 0 ∞ = 000 . . . .Then, we make connections with the Manneville-Pomeau map f : [0, 1] . In particular we show that the lift in Σ of log f is in the stable set of V * .In the second part, we characterize "good" H, such that σ 2 • H = H • σ.In the last part, we study the thermodynamic formalism for some special potentials in the stable set of V * . They are called virtual Manneville-Pomeau maps.

Suppose σ is the shift acting on Bernoulli space X = {0, 1} N , and, consider a fixed function f : X → R, under the Waters's conditions (defined in a paper in ETDS 2007). For each real value t ≥ 0 we consider the Ruelle Operator L tf . We are interested in the main eigenfunction h t of L tf , and, the main eigenmeasure ν t , for the dual operator L * tf , which we consider normalized in such way h t (0 ∞ ) = 1, and, h t d ν t = 1, ∀t > 0. We denote µ t = h t ν t the Gibbs state for the potential t f . By selection of a subaction V , when the temperature goes to zero (or, t → ∞), we mean the existence of the limitwhere h(ρ) is the Kolmogorov entropy of ρ [23].A probabilityρ which maximizes f d ρ, ρ ∈ M,