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November  2021, 41(11): 5165-5182. doi: 10.3934/dcds.2021072

Uniqueness properties of the KAM curve

IMJ-PRG, Université de Paris, Paris, France

Received  April 2020 Revised  March 2021 Published  November 2021 Early access  April 2021

Classical KAM theory guarantees the existence of a positive measure set of invariant tori for sufficiently smooth non-degenerate near-integrable systems. When seen as a function of the frequency this invariant collection of tori is called the KAM curve of the system. Restricted to analytic regularity, we obtain strong uniqueness properties for these objects. In particular, we prove that KAM curves completely characterize the underlying systems. We also show some of the dynamical implications on systems whose KAM curves share certain common features.

Citation: Frank Trujillo. Uniqueness properties of the KAM curve. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5165-5182. doi: 10.3934/dcds.2021072
References:
[1]

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L. H. EliassonB. Fayad and R. Krikorian, Around the stability of KAM tori, Duke Mathematical Journal, 164 (2015), 1733-1775.  doi: 10.1215/00127094-3120060.  Google Scholar

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J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Communications on Pure and Applied Mathematics, 35 (1982), 653-696.  doi: 10.1002/cpa.3160350504.  Google Scholar

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Z.-J. Shang, A Note on the KAM Theorem for Symplectic Mappings, Journal of Dynamics and Differential Equations, 12 (2000), 357-383.  doi: 10.1023/A:1009068425415.  Google Scholar

show all references

References:
[1]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, 3, Springer Science & Business Media, 2007.  Google Scholar

[2]

J.-B. T. Bost, Tores invariants des systèmes dynamiques hamiltoniens (d'après Kolmogorov, Arnol'd, Moser, Rüssmann, Zehnder, Herman, Pöschel), Astérisque, 113–157.  Google Scholar

[3]

A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics, 1764, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-540-45330-7.  Google Scholar

[4]

C. Carminati, S. Marmi and D. Sauzin, There is only one KAM curve, Nonlinearity, 27 (2014), 2035. doi: 10.1088/0951-7715/27/9/2035.  Google Scholar

[5]

L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals——elliptic case, Commentarii Mathematici Helvetici, 65 (1990), 4-35.  doi: 10.1007/BF02566590.  Google Scholar

[6]

L. H. EliassonB. Fayad and R. Krikorian, Around the stability of KAM tori, Duke Mathematical Journal, 164 (2015), 1733-1775.  doi: 10.1215/00127094-3120060.  Google Scholar

[7]

V. F. Lazutkin, Existence of caustics for the billiard problem in a convex domain, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 37 (1973), 186-216.   Google Scholar

[8]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Communications on Pure and Applied Mathematics, 35 (1982), 653-696.  doi: 10.1002/cpa.3160350504.  Google Scholar

[9]

Z.-J. Shang, A Note on the KAM Theorem for Symplectic Mappings, Journal of Dynamics and Differential Equations, 12 (2000), 357-383.  doi: 10.1023/A:1009068425415.  Google Scholar

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