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A KAM phenomenon for singular holomorphic vector fields

Laurent Stolovitch — 2005

Publications Mathématiques de l'IHÉS

Let X be a germ of holomorphic vector field at the origin of and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic...

Sur un théorème de Dulac

Laurent Stolovitch — 1994

Annales de l'institut Fourier

Nous considérons les champs de vecteurs analytiques de ( n , 0 ) de partie linéaire diagonale non nulle et dont les valeurs propres λ i vérifient des relations de résonances toutes engendrées par une seule relation ( r , λ ) = 0 pour un certain vecteur r n non nul. Nous montrons que, dans un système de coordonnées locales holomorphes au voisinages de 0 n , de tels champs de vecteurs se “mettent" sous une forme normale , tout en exhibant des variétés invariantes, si l’on fait une hypothèse de . Nos résultats généralisent,...

Smooth Gevrey normal forms of vector fields near a fixed point

Laurent Stolovitch — 2013

Annales de l’institut Fourier

We study germs of smooth vector fields in a neighborhood of a fixed point having an hyperbolic linear part at this point. It is well known that the “small divisors” are invisible either for the smooth linearization or normal form problem. We prove that this is completely different in the smooth Gevrey category. We prove that a germ of smooth α -Gevrey vector field with an hyperbolic linear part admits a smooth β -Gevrey transformation to a smooth β -Gevrey normal form. The Gevrey order β depends on...

Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation

Eric LombardiLaurent Stolovitch — 2010

Annales scientifiques de l'École Normale Supérieure

In this article, we study germs of holomorphic vector fields which are “higher order” perturbations of a quasihomogeneous vector field in a neighborhood of the origin of n , fixed point of the vector fields. We define a “Diophantine condition” on the quasihomogeneous initial part S which ensures that if such a perturbation of S is formally conjugate to S then it is also holomorphically conjugate to it. We study the normal form problem relatively to S . We give a condition on S that ensures that there...

Small divisors and large multipliers

Boele BraaksmaLaurent Stolovitch — 2007

Annales de l’institut Fourier

We study germs of singular holomorphic vector fields at the origin of n of which the linear part is 1 -resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some “sectorial domains” which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms....

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