On the existence of maximizing measures for irreducible countable Markov shifts: a dynamical proof

TL;DR

This paper proves the existence of maximizing measures for coercive functions with bounded variation on irreducible countable Markov shifts, extending results beyond the finitely primitive case and applicable to full shifts over positive reals.

Contribution

It provides the first proof of maximizing measures in the general irreducible non-compact setting, using a new dynamical approach.

Findings

Maximizing measures exist for coercive bounded variation functions on irreducible countable Markov shifts.

Supports of these measures are contained in finite alphabet subshifts.

The technique applies to the full shift over positive real sequences.

Abstract

We prove that if $\Sigma_{\mathbf A}(\mathbb N)$ is an irreducible Markov shift space over $\mathbb N$ and $f:\Sigma_{\mathbf A}(\mathbb N) \rightarrow \mathbb R$ is coercive with bounded variation then there exists a maximizing probability measure for f, whose support lies on a Markov subshift over a finite alphabet. Furthermore, the support of any maximizing measure is contained in this same compact subshift. To the best of our knowledge, this is the first proof beyond the finitely primitive case on the general irreducible non-compact setting. It's also noteworthy that our technique works for the full shift over positive real sequences.