João T A Gomes | Searcharxiv

arXiv2026

Dense periodic optimization for countable Markov shift via Aubry points

For transitive Markov subshifts over countable alphabets, this note ensures that a dense subclass of locally Hölder continuous potentials admits at most a single periodic probability as a maximizing measure. We resort to concepts analogous to those introduced by Mather and Mañé 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.

math.DS↗

arXiv2025

Maximizing measures for countable alphabet shifts via blur shift spaces

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.

math.DS↗