This article is cited in 13 papers

KAM-tori Near an Analytic Elliptic Fixed Point

L. Hakan Eliasson^a, Bassam Fayad^a, Raphaël Krikorian<sup>b

a</sup> Université Paris-Diderot, Institut de Mathématiques de Jussieu, UFR de Mathématiques, IMJ-PRG, Universite Paris Diderot, Batiment Sophie Germain 75205 Paris Cedex 13
^b LPMA, Université Pierre et Marie Curie, 4 pl. Jussieu, 75252 Paris Cedex 05, France

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 $\omega_0$ 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 $\omega_0$ 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 $\omega_0$.
This is an extension of previous results obtained in [1] to the case of an elliptic fixed point.

Keywords: Hamiltonian dynamics, elliptic fixed points, normal forms, KAM theory, invariant tori, Russmann’s condition, Herman’s conjecture, stability.

MSC: 34-XX, 37-XX, 34C20, 70Hxx, 37J10

Received: 02.12.2013
Accepted: 05.12.2013

Language: English

DOI: 10.1134/S1560354713060154

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