  For a finitely irreducible countable Markov shift and a potential with
summable variations, we provide a condition on the associated pressure function
which ensures that Bowen's Gibbs state, the equilibrium state, and the
minimizer of the level-2 large deviations rate function are all unique and they
coincide. From this, we deduce that all periodic points weighted with the
potential equidistribute with respect to the Gibbs-equilibrium state as the
periods tend to infinity. Applications are given to the Gauss map, and the
Bowen-Series map associated with a finitely generated free Fuchsian group with
parabolic elements.
