Zero Temperature Limits for Quotients of Potentials in Countable Markov Shifts

In this article we study ergodic optimization for quotients of functions, generalizing the classical setting in which only one function is considered. We study (non-compact) countable Markov shifts and construct a framework which allow us to prove that a sequence of equilibrium measures has an accumulation point which maximizes the integral of the quotients among the (non-compact) space of invariant measures. We provide applications of this theory to study classic ergodic optimization problems for suspension flows defined on countable Markov shifts.