Unique Equilibrium States for Geodesic Flows on Flat Surfaces with Singularities

Abstract Consider a compact surface of genus $\geq 2$ equipped with metric that is flat everywhere except at finitely many cone points angles greater than $2\pi $. Following the technique in work Burns, Climenhaga, Fisher, and Thompson, we prove sufficiently regular potential functions have unique equilibrium states if singular set does not support full pressure. Moreover, show pressure gap holds for any locally constant on neighborhood set. Finally, establish corresponding $K$-property closed geodesics equidistribute.