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Entropies of compact strictly convex projective manifolds

Abstract : Let M be a compact manifold of dimension n with a strictly convex projective structure. We consider the geodesic flow of the Hilbert metric on it, which is known to be Anosov. We prove that its topological entropy is less than n-1, with equality if and only if the structure is Riemannian, that is hyperbolic. As a corollary, we get that the volume entropy of a divisible strictly convex set is less than n-1, with equality if and only if it is an ellipsoid.
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Contributor : Mickaël Crampon Connect in order to contact the contributor
Submitted on : Thursday, April 16, 2009 - 12:22:36 PM
Last modification on : Friday, June 19, 2020 - 9:22:04 AM
Long-term archiving on: : Thursday, June 10, 2010 - 8:37:50 PM

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Mickaël Crampon. Entropies of compact strictly convex projective manifolds. Journal of modern dynamics, American Institute of Mathematical Sciences, 2009, 3 (4), pp.511-547. ⟨10.3934/jmd.2009.3.511⟩. ⟨hal-00375776⟩

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