This article is cited in 5 papers

Holomorphic Normal Form of Nonlinear Perturbations of Nilpotent Vector Fields

Laurent Stolovitch^a, Freek Verstringe<sup>b

a</sup> CNRS, Laboratoire J.-A. Dieudonné U.M.R. 6621, Université de Nice — Sophia Antipolis, Parc Valrose 06108 Nice Cedex 02, France
^b Royal Observatory of Belgium, Ringlaan 3, 1180 Brussels, Belgium

Abstract: We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension $n\geqslant 3$. 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–Żoładek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton's method and $\mathfrak{sl}_2(\mathbb C)$-representations.

Keywords: local analytic dynamics, fixed point, normal form, Belitskii normal form, small divisors, Newton method, analytic invariant manifold, complete integrability.

MSC: 34M35, 34C20, 37J40, 37F50, 58C15, 34C45

Language: English

DOI: 10.1134/S1560354716040031

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