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

Abstract Consider a topologically transitive countable Markov shift $\Sigma $ and summable locally constant potential $\phi with finite Gurevich pressure $\mathrm {Var}_1(\phi ) < \infty . We prove the existence of limit $\lim _{t \to } \mu _t$ in weak $^\star topology, where $\mu is unique equilibrium state associated to $t\phi In addition, we present examples at zero temperature exists for potentials satisfying more general conditions.