Patterson–Sullivan densities in convex projective geometry

Abstract

For any rank-one convex projective manifold with a compact convex core, we prove that there exists a unique probability measure of maximal entropy on the set of unit tangent vectors whose geodesic is contained in the convex core, and that it is mixing. We use this to establish asymptotics for the number of closed geodesics. In order to construct the measure of maximal entropy, we develop a theory of Patterson–Sullivan densities for general rank-one convex projective manifolds. In particular, we establish a Hopf–Tsuji–Sullivan–Roblin dichotomy, and prove that, when it is finite, the measure on the unit tangent bundle induced by a Patterson–Sullivan density is mixing under the action of the geodesic flow.