Research

Let S be 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. We examine the geodesic flow on S and prove local product structure for a wide class of equilibrium states. Using this, we establish the Bernoulli property for these systems. We also establish local product structure for a similar class of equilibrium states for geodesic flows on rank 1, nonpositively curved manifolds.

We show that for certain subadditive equilibrium states, the K-property implies the Bernoulli property. We do this by showing that they satisfy a version of local product structure.

Given a flat surface with finitely many conical singularities of angle greater than 2pi,  we establish uniqueness of equilibrium states for a wide class of potentials, following the technique in the work of Burns, Climenhaga, Fisher, and Thompson. We also show that these equilibrium states have the K-property.

We establish general criteria for proving the K-property, building on the Climenhaga-Thompson machinery. We introduce one-sided lambda decompositions, and apply the criteria to both Mañé diffeomorphisms and the Katok map 

We generalize Ledrappier's criterion for the K-property to subadditive equilibrium states, and use it to show that totally ergodic subadditive equilibrium states with bounded distortion and quasi-multiplicativity have the K-property. We apply this to subadditive potentials arising from certain classes of matrix cocycles.

Given a closed rank one manifold, Burns, Climenhaga, Fisher, and Thompson established uniqueness of equilibrium states for a wide class of potentials. We show that these equilibrium states have the K-property, and further, that the Bowen-Margulis measure is Bernoulli.