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arXiv · 1705.06909

KAM, $α$-Gevrey regularity and the $α$-Bruno-Rüssmann condition

Abed Bounemoura·Jacques Féjoz
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Abstract

We prove a new invariant torus theorem, for $α$-Gevrey smooth Hamiltonian systems, under an arithmetic assumption which we call the $α$-Bruno-Rüssmann condition, and which reduces to the classical Bruno-Rüssmann condition in the analytic category. Our proof is direct in the sense that, for analytic Hamiltonians, we avoid the use of complex extensions and, for non-analytic Hamiltonians, we do not use analytic approximation nor smoothing operators. Following Bessi, we also show that if a slightly weaker arithmetic condition is not satisfied, the invariant torus may be destroyed. Crucial to this work are new functional estimates in the Gevrey class.

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Abed Bounemoura, Jacques Féjoz. 2017-06-26. KAM, $α$-Gevrey regularity and the $α$-Bruno-Rüssmann condition. https://arxiv.org/abs/1705.06909

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