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arXiv · 2210.04383

KAM theorem on modulus of continuity about parameter

Zhicheng Tong·Jiayin Du·Yong Li
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Abstract

In this paper, we study the Hamiltonian systems $ H\left( {y,x,\xi ,\varepsilon } \right) = \left\langle {\omega \left( \xi \right),y} \right\rangle + \varepsilon P\left( {y,x,\xi ,\varepsilon } \right) $, where $ \omega $ and $ P $ are continuous about $ \xi $. We prove that persistent invariant tori possess the same frequency as the unperturbed tori, under certain transversality condition and weak convexity condition for the frequency mapping $ \omega $. As a direct application, we prove a KAM theorem when the perturbation $P$ holds arbitrary H\"{o}lder continuity with respect to parameter $ \xi $. The infinite dimensional case is also considered. To our knowledge, this is the first approach to the systems with the only continuity in parameter beyond H\"older's type.

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Zhicheng Tong, Jiayin Du, Yong Li. 2022-10-10. KAM theorem on modulus of continuity about parameter. https://doi.org/10.1007/s11425-022-2102-5

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