A KAM phenomenon for singular holomorphic vector fields

Details A KAM phenomenon for singular holomorphic vector fields A KAM phenomenon for singular holomorphic vector fields

2005-03-31

arxiv.org/abs/math/0503749v2

Mathematics

Dynamical Systems

62 pages, 3 figures, to appear in Publ. Math. I.H.E.S

Scientific paper

Let $X$ be a germ of holomorphic vector field at the origin of ${\bf C}^n$ and vanishing there. We assume that $X$ is a "nondegenerate" good perturbation of a singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals. We show that $X$ admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection of a polydisc with an analytic set of the form ``resonant monomials = constants". Such a biholomorphism conjugates the restriction of $X$ to one of its invariant varieties to the restriction of a linear diagonal vector field to a toric variety. Moreover, we show that the set of "frequencies" defining the invariant sets is of positive measure.

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