Ergodicity and equidistribution in Hilbert geometry

Abstract

We show that dynamical and counting results characteristic of negatively curved Riemannian geometry, or more generally CAT(−1) or rankone CAT(0) spaces, also hold for rank-one properly convex projective manifolds or orbifolds, equipped with their Hilbert metrics, admitting finite Sullivan measures built from appropriate conformal densities. In particular, this includes geometrically finite convex projective manifolds or orbifolds whose universal covers are strictly convex with C 1 boundary.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.

Convex co‐compact representations of 3‐manifold groups

“…So Weisman's boundary extension result holds for any convex co‐compact group which is relatively hyperbolic. 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 $$\Gamma \backslash \mathcal {G}_\Omega (\Gamma)$$ is mixing with respect to a natural Bowen‐Margulis measure. (3) A new class of representations : In very recent work, Weisman [50] defines a class of representations of relatively hyperbolic groups which contains both relatively Anosov representations and convex co‐compact representations.…”