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Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

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Abstract

We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension n ≥ 3. Based on Belitskii’s work, we know that such a vector field is formally conjugate to a (formal) normal form. We give a condition on that normal form which ensures that the normalizing transformation is holomorphic at the fixed point.We shall show that this sufficient condition is a nilpotent version of Bruno’s condition (A). In dimension 2, no condition is required since, according to Stróżyna–Żołladek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton’s method and sl2(C)-representations.

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Authors and Affiliations

  1. CNRS, Laboratoire J.-A. Dieudonné U.M.R. 6621, Université de Nice — Sophia Antipolis, Parc Valrose, 06108, Nice Cedex 02, France

    Laurent Stolovitch

  2. Royal Observatory of Belgium, Ringlaan 3, 1180, Brussels, Belgium

    Freek Verstringe

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  1. Laurent Stolovitch
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  2. Freek Verstringe
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Correspondence to Laurent Stolovitch.

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Research of L. Stolovitch was supported by ANR grant “ANR-10-BLAN 0102” for the project DynPDE.

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Stolovitch, L., Verstringe, F. Holomorphic normal form of nonlinear perturbations of nilpotent vector fields. Regul. Chaot. Dyn. 21, 410–436 (2016). https://doi.org/10.1134/S1560354716040031

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