Consider a topologically transitive countable Markov shift and, let f be a summable potential with bounded variation and finite Gurevic pressure. We prove that there exists an equilibrium state µ tf for each t > 1 and that there exists accumulation points for the family (µ tf )t>1 as t → ∞. We also prove that the Kolmogorov-Sinai entropy is continuous at ∞ with respect to the parameter t, that is limt→∞ h(µ tf ) = h(µ∞), where µ∞ is an accumulation point of the family (µ tf )t>1. These results do not depend on the existence of Gibbs measures and, therefore, they extend results of [12] and [16] for the existence of equilibrium states without the BIP property, [9] for the existence of accumulation points in this case and, finally, we extend completely the result of [14] for the entropy zero temperature limit beyond the finitely primitive case.

We are interested in the study of Gibbs and equilbrium probabilities on the lattice R N . Consider the unilateral full-shift defined on the noncompact set R N and an α-Hölder continuous potential A from R N into R. From a suitable class of a priori probability measures ν (over the Borelian sets of R) we define the Ruelle operator associated to A (using an adequate extension of this operator to the compact set R N = (S 1 ) N ) and we show the existence of eigenfunctions, conformal probability measures and equilibrium states associated to A. The above, can be seen as a generalization of the results obtained in the compact case for the XY-model. We also introduce an extension of the definition of entropy and show the existence of A-maximizing measures. Moreover, we prove the existence of an involution kernel for A. Finally, we build a Gibbsian specification for the Borelian sets on the set R N and we show that this family of probability measures satisfies a FKGinequality.

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Consider m ∈ N and β ∈ (1, m + 1]. Assume that a ∈ R can be represented in base β using a development in series a = ∞ n=1 x(n)β −n , where the sequence x = (x(n)) n∈N takes values in the alphabet Am := {0, . . . , m}. The above expression is called the β-expansion of a and it is not necessarily unique. We are interested in sequences x = (x(n)) n∈N ∈ A N m which are associated to all possible values a which have a unique expansion. We denote the set of such x (with some more technical restrictions) by X m,β ⊂ A N m . The space X m,β is called the symmetric β-shift associated to the pair (m, β). It is invariant by the shift map but in general it is not a subshift of finite type. Given a Hölder continuous potential A : X m,β → R, we consider the Ruelle operator L A and we show the existence of a positive eigenfunction ψ A and an eigenmeasure ρ A for some values of m and β. We also consider a variational principle of pressure. Moreover, we prove that the family of entropies (h(µ tA )) t>0 converges, when t → ∞, to the maximal value among the set of all possible values of entropy of all A-maximizing probabilities.

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