Small divisors and large multipliers

Details Small divisors and large multipliers Small divisors and large multipliers

2006-07-30

arxiv.org/abs/math/0607787v1

Mathematics

Dynamical Systems

22 pages, to appear in Annales de l'Institut Fourier

Scientific paper

We study germs of singular holomorphic vector fields at the origin of $\Bbb C^n$ of which the linear part is 1-resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some "sectorial domains" which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms. We show that the Gevrey ordrer of the latter is linked to the diophantine type of the small divisors.

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