Existence of the zero-temperature limit of equilibrium states on topologically transitive countable Markov shifts

Consider a topologically transitive countable Markov shift Sigma and a summable locally constant potential f with finite Gurevich pressure and Var1(phi) < infinity. We prove the existence of the limit lim(t ->infinity) mu(t) in the weak mu(t) topology, where mu(t) is the unique equilibrium state associated to the potential t(phi). In addition, we present examples where the limit at zero temperature exists for potentials satisfying more general conditions.