The boundary of rank-one divisible convex sets

Abstract

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

Ergodicity and equidistribution in Hilbert geometry

“…Using Zimmer's higher-rank rigidity result [72,Th. 1.4], the first author proved [10] that if M is rank-one and compact, then SM bip = SM . We will see (Corollary 3.12) that the geodesic flow is topologically mixing on the biproximal unit tangent bundle of non-elementary rank-one convex projective manifolds.…”