A Large deviations principle at zero temperature for stationary Markov equilibrium states on countable Markov shifts

Consider a topologically transitive unilateral countable Markov shift $\mathrm{\Sigma }$, a locally constant potential $\phi :\mathrm{\Sigma }\to \mathbb{R}$ satisfying suitable conditions, and assume that ${\mu }_t$ is the unique stationary Markov equilibrium state associated to the potential $t\phi$ for each $t\ge 1$. In this paper we prove a first level large deviations principle at zero temperature for the family of equilibrium states $({\mu }_t{)}_{t\ge 1}$ and we extend the result to the setting of bilateral countable Markov shifts.