Dense periodic optimization for countable Markov shift via Aubry points
Eduardo Garibaldi, João T A Gomes
TL;DR
The work develops a non-compact, countable-alphabet counterpart of Aubry–Mather ergodic optimization by introducing Mañé-type action potentials and the Peierls barrier for summable-variation potentials on transitive Markov shifts. It proves the existence of continuous sub-actions arising from Aubry points and uses them to craft perturbations that force a dense class of locally Hölder potentials to have at most one periodic maximizing measure, with stronger one-periodicity results under additional structural assumptions. Central contributions include a clear Mañé potential–Peierls barrier framework in the countable setting, a perturbative strategy to realize dense periodic-maximization phenomena, and a coercive-potential corollary that guarantees a single periodic maximizer after small perturbations. The Appendix further strengthens the conclusion under a finite cyclic predecessor condition, yielding exact uniqueness of a periodic maximizing measure. Overall, the paper extends Aubry–Mather ideas to non-compact symbolic dynamics and clarifies how periodic maximizing behavior arises densely in this broader context, with implications for the thermodynamic-like structure of ergodic optimization on countable shifts.
Abstract
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.