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

## TL;DR

This paper establishes conditions under which the equilibrium state for a countable Markov shift is unique and coincides with the minimizer of a large deviations rate function, leading to equidistribution of periodic points.

## Contribution

It provides a new criterion on the pressure function ensuring uniqueness of the Gibbs state and its coincidence with the large deviations minimizer for countable Markov shifts.

## Key findings

- Unique equilibrium state under specified pressure conditions
- Periodic points equidistribute with respect to the Gibbs state
- Applications to Gauss map and Fuchsian groups

## Abstract

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.