KAM-tori near an analytic elliptic fixed point |
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Authors: | L. Hakan Eliasson Bassam Fayad Raphaël Krikorian |
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Affiliation: | 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 2. Institut de Mathématiques de Jussieu, UFR de Mathématiques, IMJ-PRG, CNRS, Batiment Sophie Germain, 75205, Paris Cedex 13, France 3. LPMA, Université Pierre et Marie Curie, 4 pl. Jussieu, 75252, Paris Cedex 05, France
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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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