Abstract Given l > 2 ν > 2 d ≥ 4 , we prove the persistence of a Cantor–family of KAM tori of measure O ( e 1 / 2 − ν / l ) for any non–degenerate nearly integrable Hamiltonian system of class C l ( D × T d ) , where ν − 1 is the Diophantine power of the frequencies of the persistent KAM tori and D ⊂ R d is a bounded domain, provided that the size e of the perturbation is sufficiently small. This extends a result by D. Salamon in [25] according to which we do have the persistence of a single KAM torus in the same framework. Moreover, as for the persistence of a single torus, the regularity assumption is essentially optimal.

中文翻译：

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有限可微哈密顿系统的 KAM 定理

摘要 给定 l > 2 ν > 2 d ≥ 4 ，我们证明了对于任何非退化的近可积类 C l 哈密顿系统，测度为 O ( e 1 / 2 − ν / l ) 的 KAM 环的康托族的持久性( D × T d ) ，其中 ν − 1 是持久 KAM tori 频率的丢番图幂，D ⊂ R d 是有界域，前提是扰动的大小 e 足够小。这扩展了 D. Salamon 在 [25] 中的结果，根据该结果，我们确实在同一框架中具有单个 KAM 环的持久性。此外，对于单个环面的持久性，规律性假设本质上是最优的。