Large deviation principles for countable markov shifts

We establish the large deviation principle for a topological Markov shift over infinite alphabet which satisfies strong connectivity assumptions called “finite irreducibility” or “finite primitiveness”. More precisely, we assume the existence of a Gibbs state for a potential φ in the sense of Bowen, and prove the level-2 large deviation principles for the distribution of empirical means under the Gibbs state, as well as that of weighted periodic points and iterated preimages. The rate function is written with the pressure and the free energy associated with the potential φ.