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

Abstract

Let ${\Sigma }_A(\mathbb{N})$ be a topologically mixing countable Markov shift with the BIP property over the alphabet $\mathbb{N}$ and $f:{\Sigma }_A(\mathbb{N})\to \mathbb{R}$ 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 $({\mu }_{\beta }{)}_{\beta >0}$ where each ${\mu }_{\beta }$ is the Gibbs measure associated to the potential $\beta f$. 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.

This paper was submitted to arxiv without the permission of the two first authors. This is just a draft terribly written. We will submit a new and correct version in a couple of weeks