Maximizing measures for countable alphabet shifts via blur shift spaces

TL;DR

This paper establishes conditions for the existence of maximizing measures on countable alphabet shifts using blur shift spaces, extending previous methods beyond Markovian systems.

Contribution

It introduces a generalized approach employing blur shift spaces to ensure maximizing measures for broader countable alphabet shifts, not limited to Markovian cases.

Findings

Guarantees existence of maximizing measures under certain conditions.

Provides a convex characterization of blur invariant probabilities.

Ensures compactness of the set of blur invariant measures.

Abstract

For upper semi-continuous potentials defined on shifts over countable alphabets, this paper ensures sufficient conditions for the existence of a maximizing measure. We resort to the concept of blur shift, introduced by T. Almeida and M. Sobottka as a compactification method for countable alphabet shifts consisting of adding new symbols given by blurred subsets of the alphabet. Our approach extends beyond the Markovian case to encompass more general countable alphabet shifts. In particular, we guarantee a convex characterization and compactness for the set of blur invariant probabilities with respect to the discontinuous shift map.