Unique equilibrium states for geodesic flows on flat surfaces with singularities

Benjamin Call, David Constantine, Alena Erchenko, Noelle Sawyer, Grace Work

Published 2021-01-28Version 1

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