Normal form of holomorphic vector fields with an invariant torus under Brjuno’s A condition
Annales de l'Institut Fourier, Volume 66 (2016) no. 5, p. 1987-2020

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 S can be holomorphically conjugated to a normal form, i.e. a vector field which commutes with S, under two arithmetical conditions known as Brjuno’s γ and ω conditions, and an algebraic condition known as Brjuno’s A-condition, which requires the formal normal form to be proportional to S.

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 γ,ω et A de Brjuno.

Received : 2014-12-12
Revised : 2015-09-15
Accepted : 2016-02-08
Published online : 2016-07-28
DOI : https://doi.org/10.5802/aif.3055
Classification:  34A34,  34K17,  37J40,  32M25,  37F75,  37G05
Keywords: formes normales, tore invariant, condition de Brjuno, petits diviseurs, KAM, résonances
@article{AIF_2016__66_5_1987_0,
     author = {Chavaudret, Claire},
     title = {Normal form of holomorphic vector fields with an invariant torus under Brjuno's A condition},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {5},
     year = {2016},
     pages = {1987-2020},
     doi = {10.5802/aif.3055},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 66 (2016) no. 5, pp. 1987-2020. doi : 10.5802/aif.3055. http://www.numdam.org/item/AIF_2016__66_5_1987_0/

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