Hamiltonian perturbation theory for ultra-differentiable functions
Bounemoura, Abed; Féjoz, Jacques (2021), Hamiltonian perturbation theory for ultra-differentiable functions, American Mathematical Society, p. 89. 10.1090/memo/1319
Type
OuvrageLien vers un document non conservé dans cette base
https://hal.archives-ouvertes.fr/hal-01599229Date
2021Éditeur
American Mathematical Society
Isbn
978-1-4704-4691-8
Pages
89
Identifiant publication
Métadonnées
Afficher la notice complèteRésumé (EN)
Some scales of spaces of ultra-differentiable functions are introduced, having good stability properties with respect to infinitely many derivatives and compositions. They are well-suited for solving non-linear functional equations by means of hard implicit function theorems. They comprise Gevrey functions and thus, as a limiting case, analytic functions. Using majorizing series, we manage to characterize them in terms of a real sequence M bounding the growth of derivatives. In this functional setting, we prove two fundamental results of Hamiltonian perturbation theory: the invariant torus theorem, where the invariant torus remains ultra-differentiable under the assumption that its frequency satisfies some arithmetic condition which we call BR M , and which generalizes the Bruno-Rüssmann condition ; and Nekhoroshev's theorem, where the stability time depends on the ultra-differentiable class of the pertubation, through the same sequence M. Our proof uses periodic averaging, while a substitute of the analyticity width allows us to bypass analytic smoothing. We also prove converse statements on the destruction of invariant tori and on the existence of diffusing orbits with ultra-differentiable perturbations, by respectively mimicking a construction of Bessi (in the analytic category) and Marco-Sauzin (in the Gevrey non-analytic category). When the perturbation space satisfies some additional condition (we then call it matching), we manage to narrow the gap between stability hypotheses (e.g. the BR M condition) and instability hypotheses, thus circumbscribing the stability threshold. The formulas relating the growth M of derivatives of the perturbation on the one hand, and the arithmetics of robust frequencies or the stability time on the other hand, bring light to the competition between stability properties of nearly integrable systems and the distance to integrability. Due to our method of proof using width of regularity as a regularizing parameter, these formulas are closer to optimal as the the regularity tends to analyticity.Mots-clés
Hamiltonian perturbation; space of ultra-differentiable functions; KAM theory; Bruno-Rüssmann condition; Nekhoroshev theoremPublications associées
Affichage des éléments liés par titre et auteur.
-
Bounemoura, Abed (2020) Article accepté pour publication ou publié
-
Reducibility of ultra-differentiable quasi-periodic cocycles under an adapted arithmetic condition Bounemoura, Abed; Chavaudret, Claire; Liang, Shuqing (2021) Article accepté pour publication ou publié
-
Bounemoura, Abed (2016) Article accepté pour publication ou publié
-
Bounemoura, Abed; Féjoz, Jacques (2017-06) Article accepté pour publication ou publié
-
Fischler, Stephane; Bounemoura, Abed (2014) Article accepté pour publication ou publié