A Diophantine duality applied to the KAM and Nekhoroshev theorems

Abstract : In this paper, we use geometry of numbers to relate two dual Diophantine problems. This allows us to focus on simultaneous approximations rather than small linear forms. As a consequence, we develop a new approach to the perturbation theory for quasi-periodic solutions dealing only with periodic approximations and avoiding classical small divisors estimates. We obtain two results of stability, in the spirit of the KAM and Nekhoroshev theorems, in the model case of a perturbation of a constant vector field on the $n$-dimensional torus. Our first result, which is a Nekhoroshev type theorem, is the construction of a ''partial" normal form, that is a normal form with a small remainder whose size depends on the Diophantine properties of the vector. Then, assuming our vector satisfies the Bruno-Rüssmann condition, we construct an ''inverted" normal form, recovering the classical KAM theorem of Kolmogorov, Arnold and Moser for constant vector fields on torus.
Type de document :
Article dans une revue
Mathematische Zeitschrift, Springer, 2013, 275 (3), pp.1135-1167
Domaine :

https://hal.archives-ouvertes.fr/hal-00686065
Contributeur : Abed Bounemoura <>
Soumis le : mardi 19 juin 2012 - 23:44:02
Dernière modification le : dimanche 16 avril 2017 - 01:00:23
Document(s) archivé(s) le : jeudi 20 septembre 2012 - 02:46:21

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Identifiants

• HAL Id : hal-00686065, version 2
• ARXIV : 1204.1608

Citation

Abed Bounemoura, Stephane Fischler. A Diophantine duality applied to the KAM and Nekhoroshev theorems. Mathematische Zeitschrift, Springer, 2013, 275 (3), pp.1135-1167. <hal-00686065v2>

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