KAM-tori near an analytic elliptic fixed point
Résumé
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 Russmann 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.