Gibbs States On The Symbolic Space Over An In nite AlphabetR

We consider subshifts of nite type on the symbol space generated by incidence matrices over a countably innnite alphabet. We extend the deenition of topological pressure to this context and, as our main result, we construct a new class of Gibbs states of HH older continuous potentials on these symbol spaces. We establish some basic stochastic properties of these Gibbs states: exponential decay of correlations, central limit theorem and an a.s. invariance principle. This is accomplislhed via detailed studies of the associated Perron-Frobenius operator and its conjugate operator. x0. Introduction. Preliminary notation. This paper has emerged as a natural consequence of our interests in geometrical and dynamical properties of the limit sets of conformal graph directed Markov systems, a generalization of innnite conformal iterated funcion systems systematically studied in MU1], MU2] and subsequent papers. Although our paper is self-contained, it could also be considered as the rst step to developing the theory of conformal graph directed Markov systems. The central point of this paper, the existence of Gibbs states (and eigenmeasures of the operator conjugate to the Perron-Frobenius operator) for the shift map on the symbolic space generated by an innnite alphabet, and a HH older continuous potential, is contained in Section 6. This is accomplished by producing Gibbs states for symbol subspaces generated by nitely many elements of the alphabet and then demonstrating their tightness. In the rst section we generalize the concept of topological pressure to the context of symbolic space over an innnite alphabet and we provide there several variational principles. Section 2 is devoted to systematic studies of Gibbs states and their relations with equilibrium states. In Section 3 we introduce the Perron-Frobenius operator and its conjugate. We deal here with their properties and we