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

Abstract

Consider a topologically transitive unilateral countable Markov shift $\hat{I}\pounds$, a locally constant potential $\ddot{I}•:\hat{I}\pounds \to \mathbb{R}$ satisfying suitable conditions, and assume that $\hat{I}{¼}_t$ is the unique stationary Markov equilibrium state associated to the potential $t\ddot{I}•$ 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 $(\hat{I}{¼}_t{)}_{t\ge 1}$ and we extend the result to the setting of bilateral countable Markov shifts.