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

Uniqueness of minimizer for countable Markov shifts and equidistribution of periodic points