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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

The dynamics of pseudographs in convex Hamiltonian systems


Author: Patrick Bernard
Journal: J. Amer. Math. Soc. 21 (2008), 615-669
MSC (2000): Primary 37J40, 37J50
Published electronically: March 31, 2008
MathSciNet review: 2393423
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Abstract: We study the evolution, under convex Hamiltonian flows on cotangent bundles of compact manifolds, of certain distinguished subsets of the phase space. These subsets are generalizations of Lagrangian graphs, which we call pseudographs. They emerge in a natural way from Fathi's weak KAM theory. By this method, we find various orbits which connect prescribed regions of the phase space. Our study was inspired by works of John Mather. As an application, we obtain the existence of diffusion in a large class of a priori unstable systems and provide a solution to the large gap problem. We hope that our method will have applications to more examples.

RéSUMé. Nous étudions l'évolution, par le flot d'un Hamiltonien convexe sur une variété compacte, de certains ensembles de l'espace des phases. Nous appelons pseudographes ces ensembles, qui sont des généralisations de graphes Lagrangiens apparaissant de manière naturelle dans la théorie KAM faible de Fathi. Par cette méthode, nous trouvons diverses orbites qui joignent des domaines donnés de l'espace des phases. Notre étude s'inspire de travaux de John Mather. Nous obtenons l'existence de diffusion dans une large classe de systèmes à priori instables comme application de cette méthode, qui permet de résoudre le probleme de l'écart entre les tores invariants. Nous espérons que la méthode s'appliquera à d'autres exemples.


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Additional Information

Patrick Bernard
Affiliation: Université Paris-Dauphine, CEREMADE, UMR CNRS 7534, Place Marechal Lattre Tassigny, 75775 Paris, Cedex 16, France
Email: patrick.bernard@ceremade.dauphine.fr

DOI: http://dx.doi.org/10.1090/S0894-0347-08-00591-2
PII: S 0894-0347(08)00591-2
Keywords: Arnold's diffusion, Mather sets, weak KAM, Hamilton-Jacobi \nobreak equation
Received by editor(s): October 4, 2004
Published electronically: March 31, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.