Normal form of holomorphic vector fields with an invariant torus under Brjuno’s A condition
[Formes normales de champs de vecteurs holomorphes au voisinage d’un tore invariant sous la condition A de Brjuno]
Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 1987-2020.

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.

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.

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DOI : https://doi.org/10.5802/aif.3055
Classification : 34A34,  34K17,  37J40,  32M25,  37F75,  37G05
Mots clés : normal forms, invariant torus, Brjuno condition, small divisors, KAM, resonances
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     author = {Chavaudret, Claire},
     title = {Normal form of holomorphic vector fields with an invariant torus under {Brjuno{\textquoteright}s} {A} condition},
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     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3055/}
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.3055/

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