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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3055
Classification : 34A34, 34K17, 37J40, 32M25, 37F75, 37G05
Keywords: formes normales, tore invariant, condition de Brjuno, petits diviseurs, KAM, résonances
Mot clés : normal forms, invariant torus, Brjuno condition, small divisors, KAM, resonances
Chavaudret, Claire 1

1 Université de Nice Sophia-Antipolis Laboratoire J.A. Dieudonné Parc Valrose 06108 Nice Cedex 02 (France)
@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 = {https://aif.centre-mersenne.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  - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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. https://aif.centre-mersenne.org/articles/10.5802/aif.3055/

[1] Aurouet, Julien Normalisation de champs de vecteurs holomorphes et équations différentielles implicites, Université de Nice, France (2013) (Ph. D. Thesis)

[2] Brjuno, Alexander D. Analytic form of differential equations. I, Trans. Mosc. Math. Soc., Volume 25 (1971), pp. 119-262

[3] Brjuno, Alexander D. Analytic form of differential equations. II, Trans. Mosc. Math. Soc., Volume 26 (1972), pp. 199-239

[4] Bruno, Alexander D. Local methods in nonlinear differential equations, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1989, x+348 pages | DOI

[5] Giorgilli, Antonio; Marmi, Stefano Convergence radius in the Poincaré-Siegel problem, Discrete Contin. Dyn. Syst. Ser. S, Volume 3 (2010) no. 4, pp. 601-621 | DOI

[6] Lombardi, Eric; Stolovitch, Laurent Normal forms of analytic perturbations of quasihomogeneous vector fields: rigidity, invariant analytic sets and exponentially small approximation, Ann. Sci. Éc. Norm. Supér. (4), Volume 43 (2010) no. 4, pp. 659-718

[7] Meziani, Abdelhamid Normalization and solvability of vector fields near trapped orbits (to appear in Trans. Amer. Math. Soc.)

[8] Siegel, Carl Ludwig Iteration of analytic functions, Ann. of Math. (2), Volume 43 (1942), pp. 607-612 | DOI

[9] Stolovitch, Laurent Singular complete integrability, Inst. Hautes Études Sci. Publ. Math. (2000) no. 91, p. 133-210 (2001) | DOI

Cité par Sources :