Entropy rigidity and Hilbert volume | Zendy

Ilesanmi Adeboye | Zendy; Harrison Bray | Zendy; David Constantine | Zendy

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Entropy rigidity and Hilbert volume

Author(s) -

Ilesanmi Adeboye,

Harrison Bray,

David Constantine

Publication year - 2019

Publication title -

discrete and continuous dynamical systems

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.289

H-Index - 70

eISSN - 1553-5231

pISSN - 1078-0947

DOI - 10.3934/dcds.2019075

Subject(s) - mathematics , pure mathematics , bounded function , regular polygon , entropy (arrow of time) , hilbert space , geodesic , mathematical analysis , rigidity (electromagnetism) , topological entropy , hyperbolic manifold , geometry , physics , hyperbolic function , quantum mechanics

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.

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