Thermodynamic formalism for countable Markov shifts associated to non-Holder potentials.

We are going to establish the Ruelle-Perron-Frobenius theorem concerning the existence of an eigenmeasure of the R-P-F operator defined for a countable Markov shift and associated to a non-Holder potential. This result is a generalization of theorems of O. Sarig (ETDS, 1999) and R.D.Mauldin, M.Urbanski. The proof is based on a recent result of T.Szarek (Ann. of Prob., 2006) on the existence of an invariant measure for a Feller process defined on a Polish space.