Equilibrium states for geodesic flows over closed rank 1 manifolds were studied recently in Burns, Climenhaga, Fisher, and Thompson [Geom. Funct. Anal. 28 (2018), no. 5, 1209–1259]. For sufficiently regular potentials, it was shown that if the singular set does not carry full pressure, then the equilibrium state is unique. The main result of this paper is that these equilibrium states have the Kolmogorov property. In particular, these measures are mixing of all orders and have positive entropy. For the Bowen‐Margulis measure, we go further and obtain the Bernoulli property from the Kolmogorov property using classic arguments from Ornstein theory. Our argument for the Kolmogorov property is based on an idea due to Ledrappier. We prove uniqueness of equilibrium states on the product of the system with itself. To carry this out, we develop techniques for uniqueness of equilibrium states which apply in the presence of the 2‐dimensional center direction which appears for a product of flows. This is a key technical challenge of this paper.

We set out some general criteria to prove the K-property, refining the assumptions used in an earlier paper for the flow case, and introducing the analogous discrete-time result. We also introduce one-sided $\lambda $ -decompositions, as well as multiple techniques for checking the pressure gap required to show the K-property. We apply our results to the family of Mañé diffeomorphisms and the Katok map. Our argument builds on the orbit decomposition theory of Climenhaga and Thompson.

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 that 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.

By generalizing Ledrappier's criterion [Mesures d'èquilibre d'entropie complètement positive, in Systèmes dynamiques II-Varsovie, number 50 in Astérisque, Société mathématique de France, 1977, pp. 251-272] for the K-property of equilibrium states, we extend the criterion to subadditive potentials. In particular, supposing that the unique equilibrium state for a subadditive potential with quasimultiplicativity and bounded distortion is totally ergodic, we show that it has the K-property. We apply this result to subadditive potentials arising from certain classes of matrix cocycles; for the norm potentials of irreducible locally constant cocycles and the singular value potentials of typical cocycles, we show that their unique equilibrium states have the K-property. This partly generalizes the work of Morris [Ergodic properties of matrix equilibrium states, Ergodic Theory Dyn. Syst. 38(6), 2018, pp. 2295-2320] on irreducible locally constant cocycles and their subadditive equilibrium states.