Let sigma: I pound -> I pound be the left shift acting on I pound, 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 Holder continuous potential A: I pound -> a'e. 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).413321338