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 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.

Ergodicity and equidistribution in Hilbert geometry

“…All the convex projective manifolds studied in the work mentioned above, as well as in the present paper and in [9,12], satisfy the rank-one condition, which was introduced by M. Islam [42] and A. Zimmer [72].…”

“…(ii) and Theorem C follow from the work of Benoist [6] (see the end of §2.1 below) together with general equidistribution and counting results for Anosov systems due to Margulis [46]. Weaker results of a similar flavor also follow from Bray's work [17] if Ω/Γ is compact three-dimensional, from Islam's work [42] if Ω/Γ is compact, and from the first author's work [12] when Γ is convex cocompact.…”

“…In [9] and [12], the first author obtained good dynamical properties for the Hilbert geodesic flow in the setting of rank-one properly convex Hilbert geometries, including topological mixing and strong mixing of the geodesic flow with respect to a Sullivan measure (constructed via a Patterson-Sullivan density).…”

Convex co‐compact representations of 3‐manifold groups

“…In [30], we also prove a similar boundary extension result for any naive convex co-compact group which is relatively hyperbolic (recall that naive convex co-compact groups are a strictly larger class than convex co-compact groups). ( 2) Dynamics of the geodesic flow: There have been tremendous advances by Blayac [7] and Blayac-Zhu [9] in understanding the ergodic theory of the geodesic flow on convex real projective manifolds. In the context of Theorem 1.12, these results, when combined with Theorem 1.10, imply that the geodesic flow on Γ∖ Ω (Γ) is mixing with respect to a natural Bowen-Margulis measure.…”

Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groups

“…Patterson-Sullivan measures for Anosov subgroups have been extensively studied, see [18,36,12,31]. They have also been studied for relatively Anosov subgroups of the projective general linear group which preserve a properly convex domain, see [4,5,9,40,10]. More generally, Patterson-Sullivan measures can be constructed for transverse subgroups, of which Anosov subgroups, relatively Anosov subgroups, and discrete subgroups of rank one semisimple Lie groups are examples, see [15].…”