The Weak Specification Property for Geodesic Flows on Cat(-1) Spaces

We prove that the geodesic flow on a compact locally CAT(−1) space has the weak specification property, and give various applications of this property. We show that every Hölder continuous function on the space of geodesics has a unique equilibrium state, and as a result, that the BowenMargulis measure is the unique measure of maximal entropy. We establish the equidistribution of weighted periodic orbits and the large deviations principle for all such measures. For compact locally CAT(0) spaces, we give partial results, both positive and negative, on the specification property and the existence of a coding of the geodesic flow by a suspension flow over a compact shift of finite type.