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Comptes Rendus Mathématique
Volume 335, n° 4
pages 359-364 (2002)
Doi : S1631-073X(02)02484-6
Received : 4 June 2002 ;  accepted : 24 June 2002
Une condition suffisante de minimalité pour les géodésiques de la métrique de Hofer
A sufficient minimality condition for geodesics of the Hofer's metric
 

Jérôme Le Crapper
Mathématiques UFR 920, Université Paris 6, 75252 Paris cedex 05, France 

Note présentée par Charles-Michel Marle

Résumé

On donne une nouvelle condition suffisante afin qu'un Hamiltonien non autonome H sur R2n engendre une géodésique minimisante de la distance de Hofer. Cette condition porte sur le spectre des applications linéarisées de l'isotopie {φ H t } engendrée par H aux points fixes de l'isotopie. On montre de plus que si deux difféomorphismes φ 0 et φ 1 sont reliés par une telle géodésique alors la distance de Hofer entre φ 0 et φ 1 coïncide avec celle de Viterbo. Pour citer cet article : J. Le Crapper, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 359-364.

The full text of this article is available in PDF format.
Abstract

We give a new sufficient condition for a Hamiltonian H to generate a length minimizing geodesic of the Hofer's metric on the group of Hamiltonian diffeomorphisms on R2n. This condition is related to the spectra of the linearized maps of the flow {φ H t } generated by H at the fixed points of the flow. In addition we show that if φ 0 , φ 1 are two diffeomorphisms linked by such a geodesic, then the Hofer's distance between φ 0 and φ 1 is the same as Viterbo's one. To cite this article: J. Le Crapper, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 359-364.

The full text of this article is available in PDF format.


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