Entropy rigidity and Hilbert volume

Ilesanmi Adeboye; Harrison Bray; David Constantine

doi:10.3934/dcds.2019075

Article Contents

2019, Volume 39, Issue 4: 1731-1744. Doi: 10.3934/dcds.2019075

This issue Previous Article Riccati equations for linear Hamiltonian systems without controllability condition Next Article Vortex structures for some geometric flows from pseudo-Euclidean spaces

Entropy rigidity and Hilbert volume

1.
Department of Math & Comp. Sci., Wesleyan University, 265 Church Street, Middletown, CT 06459, USA
2.
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA

Received: August 2017

Revised: September 2018

Published: January 2019

The second author was supported in part by NSF RTG grant 1045119.

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Abstract

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.

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Mathematics Subject Classification: Primary: 53A20, 53C24; Secondary: 37B40, 57M50.

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References

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