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The KAM theorem on the modulus of continuity about parameters

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Abstract

In this paper, we study the Hamiltonian systems H(y, x, ξ, ε) = 〈ω(ξ),y〉+εP(y, x, ξ, ε), where ω and P are continuous about ξ. We prove that persistent invariant tori possess the same frequency as the unperturbed tori, under a certain transversality condition and a weak convexity condition for the frequency mapping ω. As a direct application, we prove a Kolmogorov-Arnold-Moser (KAM) theorem when the perturbation P holds arbitrary Hölder continuity with respect to the parameter ξ. The infinite-dimensional case is also considered. To our knowledge, this is the first approach to the systems with the only continuity in the parameter beyond Hölder’s type.

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Acknowledgements

Yong Li was supported by National Basic Research Program of China (Grant No. 2013CB834100), National Natural Science Foundation of China (Grant Nos. 12071175, 11171132 and 11571065), Project of Science and Technology Development of Jilin Province (Grant Nos. 2017C028-1 and 20190201302JC) and Natural Science Foundation of Jilin Province (Grant No. 20200201253JC). The authors thank the referees for their help suggestions and comments, which led to a significant improvement of the paper.

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Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, 130012, China

    Zhicheng Tong & Jiayin Du

  2. Institute of Mathematics, Jilin University, Changchun, 130012, China

    Yong Li

  3. School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024, China

    Yong Li

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  1. Zhicheng Tong
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  2. Jiayin Du
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  3. Yong Li
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Correspondence to Yong Li.

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Tong, Z., Du, J. & Li, Y. The KAM theorem on the modulus of continuity about parameters. Sci. China Math. 67, 577–592 (2024). https://doi.org/10.1007/s11425-022-2102-5

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  • DOI: https://doi.org/10.1007/s11425-022-2102-5

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