Weak KAM Methods and Ergodic Optimal Problems for Countable Markov Shifts

Let \(\sigma\colon \Sigma\to \Sigma\) be the left shift acting on \(\Sigma\), a one-sided Markov subshift on a countable alphabet. Our intention is to guarantee the existence of \(\sigma\)-invariant Borel probabilities that maximize the integral of a given locally Hölder continuous potential \(A\colon \Sigma\to \mathbb{R}\). Under certain conditions, we are able to show not only that \(A\)-maximizing probabilities do exist, but also that they are characterized by the fact their support lies actually in a particular Markov subshift on a finite alphabet. To that end, we make use of objects dual to maximizing measures, the so-called sub-actions (concept analogous to subsolutions of the Hamilton-Jacobi equation), and specially the calibrated sub-actions (notion similar to weak KAM solutions).