Weak KAM methods and ergodic optimal problems for countable Markov shifts

TL;DR

This paper investigates the existence and characterization of invariant measures that maximize integrals of locally H"older continuous potentials in countable Markov shifts, using weak KAM methods and sub-actions.

Contribution

It establishes conditions under which maximizing measures exist and are supported on finite subshifts, employing weak KAM theory and calibrated sub-actions.

Findings

Existence of maximizing invariant measures under certain conditions.

Maximizing measures are supported on finite Markov subshifts.

Use of weak KAM methods to analyze ergodic optimization in countable shifts.

Abstract

Let $\sigma:\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}$ be the left shift acting on $ \boldsymbol{\Sigma} $, a one-sided Markov subshift on a countable alphabet. Our intention is to guarantee the existence of $\sigma$-invariant Borel probabilities that maximize the integral of a given locally H\"older continuous potential $ A : \boldsymbol{\Sigma} \to \mathbb R $. Under certain conditions, we are able to show not only that $A$-maximizing probabilities do exist, but also that they are characterized by the fact their support lies actually in a particular Markov subshift on a finite alphabet. To that end, we make use of objects dual to maximizing measures, the so-called sub-actions (concept analogous to subsolutions of the Hamilton-Jacobi equation), and specially the calibrated sub-actions (notion similar to weak KAM solutions).