We study the simultaneous linearizability of –actions (and the corresponding -dimensional Lie algebras) defined by commuting singular vector fields in 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 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 category, the situation is completely different. We show Sternberg’s theorem for a commuting system of vector fields with a Jordan block although they do not satisfy the condition.
Nous étudions la linéarisation simultanée de –actions (et les algèbres correspondants de Lie –dimensionelles) definie par des champs de vecteurs singuliers dans 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 -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 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.
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
@article{AIF_2008__58_1_263_0, author = {Yoshino, Masafumi and Gramchev, Todor}, title = {Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having {Jordan} blocks}, journal = {Annales de l'Institut Fourier}, pages = {263--297}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {58}, number = {1}, year = {2008}, doi = {10.5802/aif.2350}, zbl = {1137.37025}, mrnumber = {2401222}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2350/} }
TY - JOUR AU - Yoshino, Masafumi AU - Gramchev, Todor TI - Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks JO - Annales de l'Institut Fourier PY - 2008 SP - 263 EP - 297 VL - 58 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2350/ DO - 10.5802/aif.2350 LA - en ID - AIF_2008__58_1_263_0 ER -
%0 Journal Article %A Yoshino, Masafumi %A Gramchev, Todor %T Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks %J Annales de l'Institut Fourier %D 2008 %P 263-297 %V 58 %N 1 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2350/ %R 10.5802/aif.2350 %G en %F AIF_2008__58_1_263_0
Yoshino, Masafumi; Gramchev, Todor. Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks. Annales de l'Institut Fourier, Volume 58 (2008) no. 1, pp. 263-297. doi : 10.5802/aif.2350. http://archive.numdam.org/articles/10.5802/aif.2350/
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