Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds

Patterson--Sullivan densities in convex projective geometry

“…The generalisation of Bray's results [Bra20a] that we are about to state is analogous to [Kni98, Th. 1.1.i], which says that any non-positively curved rank-one compact Riemannian manifold has a unique measure of maximal entropy.…”

“…Like Bray, we use methods from the non-positively curved Riemannian world, in particular inspired by Knieper and Roblin. We generalise and improve Bray's results [Bra20a], and develop more systematically the theory of Patterson-Sullivan densities in this setting.…”

“…Conformal densities were also brought to convex projective geometry, in the strictly convex and smooth case by Crampon in his PhD thesis [Cra11] and recently by F. Zhu [Zhu], and in the non-strictly convex case in dimension 3 by Bray [Bra20a]. While Zhu does not adapt Knieper's work (hence does not prove the existence and uniqueness of the measure of maximal entropy), his adaptation of Roblin's results goes further than in the present paper, enabling him for instance to obtain more precise estimates for various counting problems.…”

“…Bray [Bra20a,Bra20b] studied the geodesic flow of 3-dimensional irreducible compact convex projective manifolds M = Ω/Γ for which Ω is not necessarily strictly convex. Although the theory of Anosov flows does not apply in this setting, he managed to construct, on the unit tangent bundle, a flow-invariant ergodic measure with maximal entropy [Bra20a,Th. 1.1].…”

Geodesic flow of nonstrictly convex Hilbert geometries

“…This paper also serves as a precursor to work of the author on the Bowen-Margulis measure of maximal entropy [9]. To that end, we verify conditions of Bowen [8] for easier computability of topological entropy: Theorem 6.2.…”