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KAM-tori near an analytic elliptic fixed point

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Abstract

We study the accumulation of an elliptic fixed point of a real analytic Hamiltonian by quasi-periodic invariant tori.

We show that a fixed point with Diophantine frequency vector ω 0 is always accumulated by invariant complex analytic KAM-tori. Indeed, the following alternative holds: If the Birkhoff normal form of the Hamiltonian at the invariant point satisfies a Rüssmann transversality condition, the fixed point is accumulated by real analytic KAM-tori which cover positive Lebesgue measure in the phase space (in this part it suffices to assume that ω 0 has rationally independent coordinates). If the Birkhoff normal form is degenerate, there exists an analytic subvariety of complex dimension at least d + 1 passing through 0 that is foliated by complex analytic KAM-tori with frequency ω 0.

This is an extension of previous results obtained in [1] to the case of an elliptic fixed point.

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu, UFR de Mathématiques, IMJ-PRG, Universite Paris Diderot, Batiment Sophie Germain, Université Paris-Diderot, 75205, Paris Cedex 13, France

    L. Hakan Eliasson

  2. Institut de Mathématiques de Jussieu, UFR de Mathématiques, IMJ-PRG, CNRS, Bâtiment Sophie Germain, 75205, Paris Cedex 13, France

    Bassam Fayad

  3. LPMA, Université Pierre et Marie Curie, 4 pl. Jussieu, 75252, Paris Cedex 05, France

    Raphaël Krikorian

Authors
  1. L. Hakan Eliasson
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  2. Bassam Fayad
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  3. Raphaël Krikorian
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Corresponding author

Correspondence to L. Hakan Eliasson.

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Dedicated to our friend Alain Chenciner

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Eliasson, L.H., Fayad, B. & Krikorian, R. KAM-tori near an analytic elliptic fixed point. Regul. Chaot. Dyn. 18, 801–831 (2013). https://doi.org/10.1134/S1560354713060154

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  • DOI: https://doi.org/10.1134/S1560354713060154

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