Large Deviation Principles for countable Markov shifts

We establish the large deviation principle for a topological Markov shift over infinite alphabet which satisfies strong combinatorial assumptions called ``finite irreducibility'' or ``finite primitiveness''. More precisely, we assume the existence of a Gibbs state for a potential $\phi$ 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 pre-images. The rate function is written with the pressure and the free energy associated with the potential $\phi$.