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

Consider a compact metric space $$(M, d_M)$$ and $$X = M^{{\mathbb {N}}}$$ . We prove Ruelle’s Perron Frobenius Theorem for class of subshifts with Markovian structure introduced in da Silva et al. (Bull Braz Math Soc 45:53–72, 2014) which are defined from continuous function $$A : M \times \rightarrow {\mathbb {R}}$$ that determines the set admissible sequences. In particular, this includes finite Markov shifts models where alphabet is given by unit circle $$S^1$$ Using involution Kernel, we characterize normalized eigenfunction Ruelle operator associated to its maximal eigenvalue present an extension corresponding Gibbs state bilateral approach. From these results, existence equilibrium states accumulation points at zero temperature particular countable shifts.