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

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