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.