  Let $\sigma:\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}$ be the left shift
acting on $ \boldsymbol{\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 H\"older
continuous potential $ A : \boldsymbol{\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).
