Almost-additive thermodynamic formalism for countable Markov shifts

This paper is devoted to extend the thermodynamic formalism theory to almost-additive sequences of continuous functions defined over topologically mixing, non-compact, countable Markov shifts. Difficulties are two fold, on the one hand we have to deal with the lack of compactness of the phase space and on the other with the non-additivity of the sequence potentials. In this context, based on the work of Sarig and also on the work of Barreira, we introduce a definition of pressure. We prove that it satisfies the variational principle and hence it is good definition. Under certain combinatorial assumptions on the shift space (that of being BIP) we prove the existence and uniqueness of Gibbs measures. Applications are given, among others, to the study of maximal Lypaunov exponents of product of matrices and to obtain a formula for the Hausdorff dimension of certain geometrical constructions.