[Formes normales de champs de vecteurs holomorphes au voisinage d’un tore invariant sous la condition A de Brjuno]
On prouve l’existence d’une forme normale analytique pour certains champs de vecteurs holomorphes au voisinage d’un point fixe et d’un tore invariant. Après avoir construit une forme normale formelle, on montre que le champ de vecteurs initial peut être analytiquement normalisé sous deux conditions arithmétiques et une condition algébrique, connues comme les conditions et de Brjuno.
This article proves the existence of an analytic normal form for some holomorphic differential systems in the neighborhood of a fixed point and of an invariant torus. Once a formal normal form is constructed, one shows that the initial system with quasilinear part can be holomorphically conjugated to a normal form, i.e. a vector field which commutes with , under two arithmetical conditions known as Brjuno’s and conditions, and an algebraic condition known as Brjuno’s -condition, which requires the formal normal form to be proportional to .
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Keywords: formes normales, tore invariant, condition de Brjuno, petits diviseurs, KAM, résonances
Mot clés : normal forms, invariant torus, Brjuno condition, small divisors, KAM, resonances
@article{AIF_2016__66_5_1987_0,
author = {Chavaudret, Claire},
title = {Normal form of holomorphic vector fields with an invariant torus under {Brjuno{\textquoteright}s} {A} condition},
journal = {Annales de l'Institut Fourier},
pages = {1987--2020},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {66},
number = {5},
year = {2016},
doi = {10.5802/aif.3055},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.3055/}
}
TY - JOUR AU - Chavaudret, Claire TI - Normal form of holomorphic vector fields with an invariant torus under Brjuno’s A condition JO - Annales de l'Institut Fourier PY - 2016 SP - 1987 EP - 2020 VL - 66 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.3055/ DO - 10.5802/aif.3055 LA - en ID - AIF_2016__66_5_1987_0 ER -
%0 Journal Article %A Chavaudret, Claire %T Normal form of holomorphic vector fields with an invariant torus under Brjuno’s A condition %J Annales de l'Institut Fourier %D 2016 %P 1987-2020 %V 66 %N 5 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.3055/ %R 10.5802/aif.3055 %G en %F AIF_2016__66_5_1987_0
Chavaudret, Claire. Normal form of holomorphic vector fields with an invariant torus under Brjuno’s A condition. Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 1987-2020. doi : 10.5802/aif.3055. http://www.numdam.org/articles/10.5802/aif.3055/
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