可微哈密顿系统的一个KAM定理
A KAM Theorem for finitely differentiable Hamiltonian systems
摘要：
给定$l>2ν>2d\geq 4$，我们证明了对于$C^l(\mathscr D\times\mathbb{T}^d)$类的任何非退化近可积哈密顿系统，只要扰动的大小$\varepsilon$足够小，其中$\mathscr D\subset \mathbb{R}^d$是有界区域，康托族KAM环面的测度$O(\varepsilon^{1/2-ν/l})$的持久性.这推广了D. Salamon在\cite{salamon2004kolmogorov}中，根据它，我们在同一个框架中确实有一个KAM环面的持久性。此外，众所周知，对于单个环面的持久性，正则性假设不能改进。
英文摘要：
Given $l>2ν>2d\geq 4$, we prove the persistence of a Cantor--family of KAM tori of measure $O(\varepsilon^{1/2-ν/l})$ for any non--degenerate nearly integrable Hamiltonian system of class $C^l(\mathscr D\times\mathbb{T}^d)$, where $\mathscr D\subset \mathbb{R}^d$ is a bounded domain, provided that the size $\varepsilon$ of the perturbation is sufficiently small. This extends a result by D. Salamon in \cite{salamon2004kolmogorov} according to which we do have the persistence of a single KAM torus in the same framework. Moreover, it is well--known that, for the persistence of a single torus, the regularity assumption can not be improved.
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