Entropy rigidity and Hilbert volume

Abstract

For a closed, strictly convex projective manifold of dimension n ≥ 3 that admits a hyperbolic structure, we show that the ratio of Hilbert volume to hyperbolic volume is bounded below by a constant that depends only on dimension. We also show that for such spaces, if topological entropy of the geodesic flow goes to zero, the volume must go to infinity. These results follow from adapting Besson-Courtois-Gallot's entropy rigidity result to Hilbert geometries.

Entropy rigidity for foliations by strictly convex projective manifolds

“…We are going also to recall briefly the work by Crampon [Cra11] about Patterson-Sullivan theory for Hilbert geometries. The existence of a Patterson-Sullivan density associated to a discrete group Γ allowed Adeboye, Bray and Constantine [ABC19] to mimic the work by Besson, Curtois and Gallot [BCG95, BCG96, BCG98] and to introduce the notion of natural map. We will conclude the section with a brief overview of this theory.…”

“…Patterson-Sullivan theory for compact Hilbert manifolds. The following section is devoted to recall the main concepts about Patterson-Sullivan theory and their application to the definition of natural map introduced by Adeboye, Bray and Constantine [ABC19,BCb]. We mainly refer to [Bra,Cra11,ABC19,BCb].…”

“…Thanks to the pioneering work by Benoist [Ben00,Ben04,Ben06], the interest in the study of convex projective structures grew rapidly. Inspired by the work done by Boland and Newberger [BN01] for Finsler manifolds, Adeboye, Bray and Constantine [ABC19,BCb] proved an entropy rigidity result for strictly convex projective manifolds with finite volume. They showed that the volume entropy rescaled by the eccentricity factor associated to the strictly convex projective structure attains its minimum value if and only if the structure is actually hyperbolic.…”

“…The proof of the above theorem is quite easy since it relies on the construction of natural maps given by Adeboye, Bray and Constantine [ABC19]. Indeed it is sufficient to construct a natural map for every leaf of F N and then glue all the natural maps together to get a global foliation-preserving map.…”

Equivariant maps for measurable cocycles with values into higher rank Lie groups

“…Among the other possible applications of the barycenter construction and natural maps, it is worth mentioning the rigidity result obtained by Boland-Connell [BC02] for foliations of Riemannian manifolds with negatively curved leaves and the rigidity phenomena proved by Boland-Newberger [BN01] and by Adeboye-Bray-Constantine [ABC19] for Finsler/Benoist manifolds. To conclude this historical introduction, we recall also the work of Lafont-Schmidt [LS06].…”