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.

💡 Research Summary

The paper investigates ergodic optimization for transitive countable‑state Markov shifts, extending results that were previously known only for compact (finite‑alphabet) systems. Let Σ be the one‑sided shift space over the countable alphabet ℕ₀ equipped with the usual metric d(x,y)=λ^{ℓ} where ℓ is the first coordinate at which x and y differ, and let σ:Σ→Σ be the left shift. A potential A:Σ→ℝ is assumed continuous and bounded above. The central object is the maximizing value
\