Regular and Chaotic Dynamics

, Volume 21, Issue 4, pp 410–436 | Cite as

Holomorphic normal form of nonlinear perturbations of nilpotent vector fields



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


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

MSC2010 numbers

34M35 34C20 37J40 37F50 58C15 34C45 


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Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.CNRS, Laboratoire J.-A. Dieudonné U.M.R. 6621Université de Nice — Sophia Antipolis, Parc ValroseNice Cedex 02France
  2. 2.Royal Observatory of BelgiumBrusselsBelgium

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