Resumen de Existence of Gibbs States and Maximizing Measures on a General One-Dimensional Lattice System with Markovian Structure

Rafael Rigão Souza, Victor Vargas

• Consider a compact metric space (M,dM) and X=MN. We prove a Ruelle’s Perron Frobenius Theorem for a class of compact subshifts with Markovian structure introduced in da Silva et al. (Bull Braz Math Soc 45:53–72, 2014) which are defined from a continuous function A:M×M→R that determines the set of admissible sequences. In particular, this class of subshifts includes the finite Markov shifts and models where the alphabet is given by the unit circle S1. Using the involution Kernel, we characterize the normalized eigenfunction of the Ruelle operator associated to its maximal eigenvalue and present an extension of its corresponding Gibbs state to the bilateral approach. From these results, we prove existence of equilibrium states and accumulation points at zero temperature in a particular class of countable Markov shifts.