  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.
