Unique equilibrium states for geodesic flows on flat surfaces with singularities

TL;DR

This paper proves the uniqueness of equilibrium states for geodesic flows on flat surfaces with cone singularities, under certain regularity and pressure gap conditions, and shows these states have strong ergodic properties.

Contribution

It extends thermodynamic formalism to flat surfaces with singularities, establishing uniqueness, ergodic properties, and equidistribution results for equilibrium states.

Findings

Unique equilibrium states exist under specified conditions.

Equilibrium states have the K-property, indicating strong ergodic behavior.

Closed geodesics equidistribute with respect to the equilibrium states.

Abstract

Consider a compact surface of genus $\geq 2$ 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 have the $K$-property, and closed regular geodesics equidistribute.