Entropy Rigidity and Hilbert Volume

TL;DR

This paper extends entropy rigidity results to convex projective manifolds, establishing volume bounds related to hyperbolic structures and analyzing the behavior of volume as entropy approaches zero.

Contribution

It adapts Besson–Courtois–Gallot's entropy rigidity to Hilbert geometries, providing new volume bounds and entropy-volume relationships for convex projective manifolds.

Findings

Hilbert volume to hyperbolic volume ratio is bounded below by a dimension-dependent constant

If entropy tends to zero, the volume tends to infinity for these manifolds

Results generalize entropy rigidity to convex projective geometries

Abstract

For a closed, strictly convex projective manifold of dimension $n\geq 3$ 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.