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.

中文翻译：

---

可数马尔可夫位移和周期点等分布的极小化器的唯一性

对于有限不可约可数马尔可夫位移和具有可求和变化的势能，我们提供了相关压力函数的条件，以确保 Bowen's Gibbs 状态、平衡状态和 2 级大偏差率函数的最小值都是唯一的，并且他们重合。由此，我们推导出所有用势均等分布加权的周期点都相对于吉布斯平衡状态，因为周期趋于无穷大。给出了高斯图的应用，以及与具有抛物线元素的有限生成的自由 Fuchsian 群相关联的 Bowen 级数图。