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A Large Deviation Principle for Gibbs States on Markov Shifts at Zero Temperature

Let $Σ_{A}(\mathbb{N})$ be a topologically mixing countable Markov shift with the BIP property over the alphabet $\mathbb{N}$ and $f: Σ_{A}(\mathbb{N}) \rightarrow \mathbb{R}$ a potential satisfying the Walters condition with finite Gurevich pressure. Under suitable hypotheses, we prove the existence of a Large Deviation Principle for the family $(μ_β)_{β> 0}$ where each $μ_β$ is the Gibbs measure associated to the potential $βf$. Our main theorem generalizes from finite to countable alphabets and also to a larger class of potentials a previous result of A. Baraviera, A. O. Lopes and P. Thieullen.