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Laurent Stolovitch

Lab. Dieudonné, Parc Valrose, 06108, Nice cedex 2
CNRS and Université de Nice Sophia Antipolis


Stolovitch L., Verstringe F.
Holomorphic Normal Form of Nonlinear Perturbations of Nilpotent Vector Fields
2016, vol. 21, no. 4, pp.  410-436
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
Citation: Stolovitch L., Verstringe F.,  Holomorphic Normal Form of Nonlinear Perturbations of Nilpotent Vector Fields, Regular and Chaotic Dynamics, 2016, vol. 21, no. 4, pp. 410-436

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