Unique equilibrium states for geodesic flows on flat surfaces with singularities

Abstract

Consider a compact surface of genus ≥ 2 equipped with a metric that is flat everywhere except at finitely many cone points with angles greater than 2π. 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.1.1. Outline of the argument. A general scheme for proving that unique equilibrium states exist was developed by Climenhaga and Thompson in [CT16], building on ideas of Bowen in [Bow75] which were extended to flows in [Fra77]. To prove that there are unique equilibrium states for a flow {f t } and a potential φ on a compact metric space X, Climenhaga and Thompson ask for the following (see [CT16, Theorems A & C]):• The pressure of obstructions to expansivity, P ⊥ exp (φ), is smaller than P (φ), and • There are three collections of orbit segments P, G, S, that we call collections of prefixes, good orbit segments, and suffixes, respectively, such that each orbit segment can be decomposed into a prefix, a good part, and a suffix (see [BCFT18, Definition 2.3]), satisfying