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Article Dans Une Revue Regular and Chaotic Dynamics Année : 2016

Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

Résumé

We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension $n \geq 3$. 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 ensure 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 Strózyna- Zoladek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton's method and $\frak{sl}_2(\Bbb C)$-representations.
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Dates et versions

hal-01336282 , version 1 (22-06-2016)

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Citer

Laurent Stolovitch, Freek Verstringe. Holomorphic normal form of nonlinear perturbations of nilpotent vector fields. Regular and Chaotic Dynamics, 2016, 21, pp.410-436. ⟨10.1134/S1560354716040031⟩. ⟨hal-01336282⟩
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