Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks

Yoshino, Masafumi; Gramchev, Todor

Annales de l'Institut Fourier, Tome 58 (2008) no. 1, pp. 263-297.

Nous étudions la linéarisation simultanée de $d$–actions (et les algèbres correspondants de Lie $d$–dimensionelles) definie par des champs de vecteurs singuliers dans ${ℂ}^n$ fixant l’origine avec des parties linéaires ayant des blocs de Jordan. Nous montrons la convergence analytique des transformations linéarisantes formelles sous une condition d’invariance géométrique pour le spectre de $d$-champs de vecteurs qui engendrent une algèbre de Lie. Si la condition n’est pas satisfaite et si il y a des petits diviseurs, nous montrons l’existence de solutions divergentes pour un système sous déterminé d’équations linéarisées homologiques. Dans le cadre de fonctions ${ℂ}^{\infty }$ la situation est complètement différente. Nous montrons le théorème de Sternberg pour une famille commutative de champs de vecteurs qui ne satisfait pas la condition.

We study the simultaneous linearizability of $d$–actions (and the corresponding $d$-dimensional Lie algebras) defined by commuting singular vector fields in ${ℂ}^n$ fixing the origin with nontrivial Jordan blocks in the linear parts. We prove the analytic convergence of the formal linearizing transformations under a certain invariant geometric condition for the spectrum of $d$ vector fields generating a Lie algebra. If the condition fails and if we consider the situation where small denominators occur, then we show the existence of divergent solutions of an overdetermined system of linearized homological equations. In the ${ℂ}^{\infty }$ category, the situation is completely different. We show Sternberg’s theorem for a commuting system of ${ℂ}^{\infty }$ vector fields with a Jordan block although they do not satisfy the condition.

Classification : 32M25, 37F50, 37G05
Keywords: singular vector field, linearization, Jordan block, homological equation, Diophantine conditions, Gevrey spaces, decomposition
Mot clés : champ de vecteurs singulier, linéarisation, bloc de Jordan, équations omologiques, conditions diophantiennes, espaces de Gevrey, décomposition

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