  We show that dynamical and counting results characteristic of
negatively-curved Riemannian geometry, or more generally CAT($-1$) or rank-one
CAT($0$) spaces, also hold for rank-one properly convex projective structures,
equipped with their Hilbert metrics, admitting finite Sullivan measures built
from appropriate conformal densities. In particular, this includes
geometrically finite convex projective structures.
  More specifically, with respect to the Sullivan measure, the Hilbert geodesic
flow is strongly mixing, and orbits and primitive closed geodesics
equidistribute, allowing us to asymptotically enumerate these objects.
