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Small divisors and large multipliers. (English) Zbl 1138.37028
Summary: We study germs of singular holomorphic vector fields at the origin of \(\mathbb C^n\) 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 order of the latter is linked to the Diophantine type of the small divisors.

MSC:
37F75 Dynamical aspects of holomorphic foliations and vector fields
34M30 Asymptotics and summation methods for ordinary differential equations in the complex domain
32S65 Singularities of holomorphic vector fields and foliations
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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