Entropy rigidity and Hilbert volume

Ilesanmi Adeboye, Harrison Bray, David Constantine
2019 Discrete and Continuous Dynamical Systems. Series A  
For a closed, strictly convex projective manifold 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.
doi:10.3934/dcds.2019075 fatcat:wjnx5wzc6nelbbv7ot5eipqaky