Pressure and equilibrium states for countable state markov shifts

We give a general definition of the topological pressureP _top (f, S) for continuous real valued functionsf: X→ℝ on transitive countable state Markov shifts (X, S). A variational principle holds for functions satisfying a mild distortion property. We introduce a new notion of Z-recurrent functions. Given any such functionf, we show a general method how to obtain tight sequences of invariant probability measures supported on periodic points such that a weak accumulation pointμ is an equilibrium state forf if and only if εf ⁻ dμ<∞. We discuss some conditions that ensure this integrability. As an application we obtain the Gauss measure as a weak limit of measures supported on periodic points.