Uniqueness properties of the KAM curve

Frank Trujillo

doi:10.3934/dcds.2021072

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2021, Volume 41, Issue 11: 5165-5182. Doi: 10.3934/dcds.2021072

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Uniqueness properties of the KAM curve

Frank Trujillo

IMJ-PRG, Université de Paris, Paris, France

Received: April 2020

Revised: March 2021

Early access: April 2021

Published: November 2021

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Abstract

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.

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Mathematics Subject Classification: 37E40, 37J40, 70H08.

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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.
[2]	J.-B. T. Bost, Tores invariants des systèmes dynamiques hamiltoniens (d'après Kolmogorov, Arnol'd, Moser, Rüssmann, Zehnder, Herman, Pöschel), Astérisque, 113–157.
[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.
[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.
[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.
[6]	L. H. Eliasson, B. Fayad and R. Krikorian, Around the stability of KAM tori, Duke Mathematical Journal, 164 (2015), 1733-1775. doi: 10.1215/00127094-3120060.
[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.
[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.
[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.