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We introduce a natural subset of the unit tangent bundle of an irreducible convex projective manifold, which is closed and invariant under the geodesic flow, and we prove that the geodesic flow is topologically mixing on it. We also show that, for higher-rank compact convex projective manifolds, the geodesic flow is topologically mixing on each connected component of the nonwandering set.

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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 rankone 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.

We prove that for any non-symmetric irreducible divisible convex set, the proximal limit set is the full projective boundary.If Ω is symmetric, then it naturally identifies with the Riemannian symmetric space of Aut(Ω), and there is yet another natural dichotomy: namely, either Aut(Ω) has real rank 1, in which case Ω is an ellipsoid and Aut(Ω) is isomorphic to PO(n, 1) for n = dim(V ) − 1, or Aut(Ω) has real rank greater than one, it is isomorphic to PGL(n, K) for some n ≥ 3, and for K = R, C, or the classical division algebra of quaternions, or of octonions if n = 3 (see for instance [Ben08, §2.4]).Recently, A. Zimmer proved the following higher-rank rigidity result [Zim, Th. 1.4], analogous to a celebrated result in Riemannian geometry by Ballmann [Bal85] and Burns-Spatzier [BS87].If Ω is not symmetric, then it is rank-one in the following sense.Definition 1.1. A divisible convex set Ω ⊂ P(V ) is said to be rank-one if there exists in ∂Ω a strongly extremal point, namely a point ξ ∈ ∂Ω such that [ξ, η]∩Ω is non-empty for any η ∈ ∂Ω {ξ} (in other words, ξ is "visible" from any other point of the projective boundary).The notion of rank-one divisible convex sets (and more generally of rank-one geodesics, automorphisms, groups of automorphisms, quotients of properly convex open sets, which we do not define