Dense periodic optimization for countable Markov shift via Aubry points

TL;DR

This paper studies the uniqueness of maximizing measures for certain countable Markov shifts, introducing Aubry points and sub-actions to establish conditions for a dense class of potentials.

Contribution

It extends concepts from Lagrangian dynamics to countable Markov shifts, proving the existence of sub-actions and the generic uniqueness of maximizing measures.

Findings

A dense subclass of potentials admits at most one maximizing measure.

Existence of continuous sub-actions in the presence of Aubry points.

Application of Aubry-Mather theory to countable Markov shifts.

Abstract

For transitive Markov subshifts over countable alphabets, this note ensures that a dense subclass of locally H\"older continuous potentials admits at most a single periodic probability as a maximizing measure. We resort to concepts analogous to those introduced by Mather and Ma\~n\'e in the study of globally minimizing curves in Lagrangian dynamics. In particular, given a summable variation potential, we show the existence of a continuous sub-action in the presence of an Aubry point.