Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

Abstract

We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension $n\ge 3$. 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 Str{ó}zyna- Zoladek, 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.

To appear in Regular and Chaotic Dynamics