Stolovitch, L ; Verstringe, F

accepted to be published in Regular and chaotic dynamics, 21 issue 4, pp. 410-436 (2016)

Abstract: We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension 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 Strozyna – Zoladek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton’s method and sl2(C)-representations.

Keyword(s): local analytic dynamics, fixed point, normal form, Belitskii normal form, small divisors, Newton method, analytic invariant manifold, complete integrability
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