Weak KAM methods and ergodic optimal problems for countable Markov shifts

Let $σ:\boldsymbolΣ\to\boldsymbolΣ$ be the left shift acting on $\boldsymbolΣ$, a one-sided Markov subshift on a countable alphabet. Our intention is to guarantee the existence of $σ$-invariant Borel probabilities that maximize the integral of a given locally Hölder continuous potential $A : \boldsymbolΣ \to \mathbb R$. 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).

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