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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Abstract. We study the hydrodynamic and the hydrostatic behavior of the Simple Symmetric Exclusion Process with slow boundary. The term slow boundary means that particles can be born or die at the boundary sites, at a rate proportional to N −θ , where θ > 0 and N is the scaling parameter. In the bulk, the particles exchange rate is equal to 1. In the hydrostatic scenario, we obtain three different linear profiles, depending on the value of the parameter θ; in the hydrodynamic scenario, we obtain that the time evolution of the spatial density of particles, in the diffusive scaling, is given by the weak solution of the heat equation, with boundary conditions that depend on θ. If θ ∈ (0, 1), we get Dirichlet boundary conditions, (which is the same behavior if θ = 0, see [7]); if θ = 1, we get Robin boundary conditions; and, if θ ∈ (1, ∞), we get Neumann boundary conditions.

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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