no. 5
Integrable analytic vector fields with a nilpotent linear part
Annales de l'Institut Fourier, Tome 45 (1995) no. 5, pp. 1449-1470.

On étudie la normalisation des champs de vecteurs analytiques à partie linéaire nilpotente. On démontre qu’un tel champ de vecteurs analytique peut être transformé en une certaine forme par des transformations convergentes s’il a une intégrale formelle non singulière. Alors on montre qu’il existe des applications analytiques paraboliques différentiablement linéarisables qui ne peuvent être plongées dans le flot d’aucun champ de vecteurs analytique avec une partie linéaire nilpotente.

We study the normalization of analytic vector fields with a nilpotent linear part. We prove that such an analytic vector field can be transformed into a certain form by convergent transformations when it has a non-singular formal integral. We then prove that there are smoothly linearizable parabolic analytic transformations which cannot be embedded into the flows of any analytic vector fields with a nilpotent linear part.

@article{AIF_1995__45_5_1449_0,
     author = {Gong, Xianghong},
     title = {Integrable analytic vector fields with a nilpotent linear part},
     journal = {Annales de l'Institut Fourier},
     pages = {1449--1470},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {45},
     number = {5},
     year = {1995},
     doi = {10.5802/aif.1502},
     mrnumber = {96m:58229},
     zbl = {0835.58032},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1502/}
}
TY  - JOUR
AU  - Gong, Xianghong
TI  - Integrable analytic vector fields with a nilpotent linear part
JO  - Annales de l'Institut Fourier
PY  - 1995
SP  - 1449
EP  - 1470
VL  - 45
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1502/
DO  - 10.5802/aif.1502
LA  - en
ID  - AIF_1995__45_5_1449_0
ER  - 
%0 Journal Article
%A Gong, Xianghong
%T Integrable analytic vector fields with a nilpotent linear part
%J Annales de l'Institut Fourier
%D 1995
%P 1449-1470
%V 45
%N 5
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1502/
%R 10.5802/aif.1502
%G en
%F AIF_1995__45_5_1449_0
Gong, Xianghong. Integrable analytic vector fields with a nilpotent linear part. Annales de l'Institut Fourier, Tome 45 (1995) no. 5, pp. 1449-1470. doi : 10.5802/aif.1502. http://www.numdam.org/articles/10.5802/aif.1502/

[1] V.I. Arnol'D and Yu. S. Il'Yashenko, Ordinary differential equations, in “Dynamical Systems I, EMS” vol. 1, Springer-Verlag, Berlin, 1990. | Zbl

[2] A. Baider and J. C. Sanders, Further reduction of the Takens-Bogdanov normal form, J. Diff. Equations, 99 (1992), 205-244. | MR | Zbl

[3] R.I. Bogdanov, Versal deformation of a singularity of a vector field on the plane in the case of zero eigenvalues, Seminar Petrovski (1976), and Selecta Math. Soviet, n° 4, 1 (1981), 389-421. | Zbl

[4] D. Cerveau and R. Moussu, Groupes d'automorphismes de (C, 0) et équations différentielles y dy + ... = 0, Bull. Soc. Math. France, 116 (1988), 459-488. | Numdam | MR | Zbl

[5] X. Gong, Divergence for the normalization of real analytic glancing hypersurfaces, Commun. Partial Diff. Equations, 19, n° 3 & 4 (1994), 643-654. | MR | Zbl

[6] R.B. Melrose, Equivalence of glancing hypersurfaces, Invent. Math., 37 (1976), 165-191. | MR | Zbl

[7] F. Takens, Singularities of vector fields, Publ. Math. I.H.E.S., 43 (1974), 47-100. | Numdam | MR | Zbl

[8] S.M. Webster, Holomorphic symplectic normalization of a real function, Ann. Scuola Norm. Sup. di Pisa, 19 (1992), 69-86. | Numdam | MR | Zbl

Cité par Sources :