Smooth linearization of germs of R 2 -actions and holomorphic vector fields
Annales de l'Institut Fourier, Tome 30 (1980) no. 1, pp. 31-64.

Cet article contient une condition générique permettant la linéarisation en classe 𝒞 k , 0k, des germes d’actions infinitésimales singulières de R 2 sur R n (n2) et de champs de vecteurs holomorphes singuliers sur C n (n1). Cela généralise un résultat de S. Sternberg pour les germes de champs de vecteurs réels, singuliers, sur R n .

The paper contains a generic condition permitting the linearization in class 𝒞 k , 0k, of germs of singular infinitesimal R 2 -actions on R n (n2) and of singular holomorphic vector fields on C n (n1). It generalizes a similar result of S. Sternberg for germs of singular (real) vector fields on R n .

@article{AIF_1980__30_1_31_0,
     author = {Dumortier, F. and Roussarie, Robert},
     title = {Smooth linearization of germs of $R^2$-actions and holomorphic vector fields},
     journal = {Annales de l'Institut Fourier},
     pages = {31--64},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {30},
     number = {1},
     year = {1980},
     doi = {10.5802/aif.774},
     zbl = {0418.58015},
     mrnumber = {81k:58060},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.774/}
}
TY  - JOUR
AU  - Dumortier, F.
AU  - Roussarie, Robert
TI  - Smooth linearization of germs of $R^2$-actions and holomorphic vector fields
JO  - Annales de l'Institut Fourier
PY  - 1980
SP  - 31
EP  - 64
VL  - 30
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.774/
DO  - 10.5802/aif.774
LA  - en
ID  - AIF_1980__30_1_31_0
ER  - 
%0 Journal Article
%A Dumortier, F.
%A Roussarie, Robert
%T Smooth linearization of germs of $R^2$-actions and holomorphic vector fields
%J Annales de l'Institut Fourier
%D 1980
%P 31-64
%V 30
%N 1
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.774/
%R 10.5802/aif.774
%G en
%F AIF_1980__30_1_31_0
Dumortier, F.; Roussarie, Robert. Smooth linearization of germs of $R^2$-actions and holomorphic vector fields. Annales de l'Institut Fourier, Tome 30 (1980) no. 1, pp. 31-64. doi : 10.5802/aif.774. https://aif.centre-mersenne.org/articles/10.5802/aif.774/

[1] A.D. Brjuno, The analytic form of differential equations, Trans. Moscow Math. Soc., Vol. 25 (1971), 131-288. | MR | Zbl

[2] C. Camacho, On Rk x Zl-actions, Dynamical Systems, Proc. of the Salvador Symposium, Ed. M.M. Peixoto, (1973), 23-70. | Zbl

[3] C. Camacho, N.H. Kuiper, J. Palis, La topologie du feuilletage d'un champ de vecteurs holomorphes près d'une singularité, C.R. Acad. Sc. Paris, t. 282 (1976), 959-961. | MR | Zbl

[4] C. Camacho, N.H. Kuiper, J. Palis, The topology of holomorphic flows with singularity, Preprint of the I.H.E.S., Bures-sur-Yvette. | Numdam | Zbl

[5] F. Dumortier, R. Roussarie, Linéarisation différentiable des germes singuliers d'actions de R2 et de champs de vecteurs holomorphes, C.R. Acad. Sc. Paris, t. 285 Série A (1977), 841. | MR | Zbl

[6] J. Guckenheimer, Hartman's theorem for complex flows in the Poincaré domain, Compositio Math., 24 (1972), 75-82. | Numdam | MR | Zbl

[7] P. Hartman, On the local linearization of differential equations, Proc. Amer. Math. Soc., 14 (1963), 568-573. | MR | Zbl

[8] M. Hirsch, C.C. Pugh, M. Shub, Invariant manifolds, Lecture Notes in Math., 583 (1977). | MR | Zbl

[9] N. Kopell, Commuting diffeomorphisms global analysis, Proc. of Symp. in Pure Math., XIV, (1970), 165-184. | MR | Zbl

[10] R. Roussarie, Modèles locaux de champs et de formes, Astérisque, Vol. 30 (1975). | MR | Zbl

[11] R. Roussarie, Singularités des germes de champs de vecteurs, Bol. Soc. Brasileira de Matemática (to appear). | Zbl

[12] G.L. Siegel, Nachr. Akad. Wiss Göttingen, Math. Phys., I (1952), 21-30. | Zbl

[13] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967). | Zbl

[14] S. Sternberg, On the structure of local homeomorphisms of Euclidean n-space II, J. of Math., Vol. 80 (1958), 623-631. | MR | Zbl

Cité par Sources :