Dynamics and entropies of Hilbert metrics

Abstract : We study the geodesic flow of a Hilbert geometry defined by a strictly convex open set with $C^1$ boundary. We get interested in its local behaviour around one specific orbit as well as its global properties on a quotient manifold. We explain why this flow has hyperbolic-like properties, by studying in particular its Lyapunov exponents, which are linked in a precise way to the shape of the boundary of the convex. We prove an entropy rigidity result for compact quotients. We also develop general tools that can be used when considering noncompact ones, following ideas and results of negative curvature. The case of geometrically finite surfaces is studied in details, and the entropy rigidity theorem is extended to finite volume surfaces.
Document type :
Theses
Mathematics [math]. Université de Strasbourg; Ruhr-Universität Bochum, 2011. English


https://tel.archives-ouvertes.fr/tel-00570002
Contributor : Mickaël Crampon <>
Submitted on : Wednesday, March 2, 2011 - 3:53:39 PM
Last modification on : Wednesday, March 2, 2011 - 4:49:10 PM

Identifiers

  • HAL Id : tel-00570002, version 2

Collections

Citation

Mickaël Crampon. Dynamics and entropies of Hilbert metrics. Mathematics [math]. Université de Strasbourg; Ruhr-Universität Bochum, 2011. English. <tel-00570002v2>

Export

Share

Metrics

Consultation de
la notice

184

Téléchargement du document

72