Hidden Gibbs measures on shift spaces over countable alphabets

TL;DR

This paper extends thermodynamic formalism and Gibbs measure theory to subshifts over countable alphabets, establishing variational principles and conditions for unique equilibrium states.

Contribution

It introduces new results on the existence and uniqueness of Gibbs measures and equilibrium states for subshifts over countable alphabets, expanding finite alphabet theories.

Findings

Proved the variational principle for topological pressure in this setting

Established conditions for existence and uniqueness of Gibbs measures

Extended finite alphabet Gibbs measure theory to countable alphabets

Abstract

We study the thermodynamic formalism for particular types of sub-additive sequences on a class of subshifts over countable alphabets. The subshifts we consider include factors of irreducible countable Markov shifts under certain conditions. We show the variational principle for topological pressure. We also study conditions for the existence and uniqueness of invariant ergodic Gibbs measures and the uniqueness of equilibrium states. As an application, we extend the theory of factors of (generalized) Gibbs measures on subshifts on finite alphabets to that on certain subshifts over countable alphabets.