Bernoullicity of equilibrium measures on countable Markov shifts

TL;DR

This paper proves that equilibrium measures for certain countable Markov shifts are Bernoulli up to a period, generalizing key theorems and establishing a variational principle and uniqueness of equilibrium measures.

Contribution

It extends the theory of equilibrium measures on countable Markov shifts by proving Bernoullicity, a variational principle, and the uniqueness of equilibrium measures under Walters conditions.

Findings

Equilibrium measures are Bernoulli up to a period.

Established a variational principle for countable Markov shifts.

Proved the uniqueness of equilibrium measures under Walters conditions.

Abstract

We study the equilibrium behaviour of a two-sided topological Markov shift with a countable number of states. We assume the potential associated with this shift is Walters with finite first variation and that the shift is topologically transitive. We show the equilibrium measure of the system is Bernoulli up to a period. In the process we generalize several theorems on countable Markov shifts. We prove a variational principle and the uniqueness of equilibrium measures. A key step is to show that functions with Walters property on a two-sided shift are cohomologous to one-sided functions with the Walters property. Then we turn to show that functions with summable variations on two-sided CMS are cohomologous to one-sided functions, also with summable variations.