Geodesic
Separatrices for non solvable dynamics on ℂ,0
Annales de l'Institut Fourier, Tome 44 (1994) no. 2, pp. 569-599.

Voir la notice de l'article provenant de la source Numdam

We define the separatrices for pseudogroups of diffeomorphisms of open neighbourhoods of the origin in the complex plane ℂ and prove their existence for non solvable pseudogroups (Theorem 1). This extends a result by Shcherbakov (in [21]) accurately. Our method also applies to prove the topological rigidity theorem for generic pseudogroups attributed to Shcherbakov (dans [20]).

Nous définissons les séparatrices pour les pseudo-groupes de difféomorphismes de voisinages ouverts de l’origine du plan complexe ℂ, et nous démontrons leur existence pour les pseudo-groupes non résolubles (Théorème 1). Ceci précise un résultat de Shcherbakov (dans [21]). Notre méthode permet aussi de démontrer le théorème de rigidité topologique pour les pseudo-groupes génériques attribué à Shcherbakov (dans [20]).

@article{AIF_1994__44_2_569_0,
     author = {Nakai, Isao},
     title = {Separatrices for non solvable dynamics on ${\mathbb {C}},0$},
     journal = {Annales de l'Institut Fourier},
     pages = {569--599},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {44},
     number = {2},
     year = {1994},
     doi = {10.5802/aif.1410},
     mrnumber = {95j:58124},
     zbl = {0804.57022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5802/aif.1410/}
}
TY  - JOUR
AU  - Nakai, Isao
TI  - Separatrices for non solvable dynamics on ${\mathbb {C}},0$
JO  - Annales de l'Institut Fourier
PY  - 1994
SP  - 569
EP  - 599
VL  - 44
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://geodesic.mathdoc.fr/articles/10.5802/aif.1410/
DO  - 10.5802/aif.1410
LA  - en
ID  - AIF_1994__44_2_569_0
ER  - 
%0 Journal Article
%A Nakai, Isao
%T Separatrices for non solvable dynamics on ${\mathbb {C}},0$
%J Annales de l'Institut Fourier
%D 1994
%P 569-599
%V 44
%N 2
%I Association des Annales de l’institut Fourier
%U http://geodesic.mathdoc.fr/articles/10.5802/aif.1410/
%R 10.5802/aif.1410
%G en
%F AIF_1994__44_2_569_0
Nakai, Isao. Separatrices for non solvable dynamics on ${\mathbb {C}},0$. Annales de l'Institut Fourier, Tome 44 (1994) no. 2, pp. 569-599. doi : 10.5802/aif.1410. http://geodesic.mathdoc.fr/articles/10.5802/aif.1410/

[1]I.N. Baker, Fractional iteration near a fixpoint of multiplier 1, J. Australian Math. Soc., 4 (1964), 143-148. | Zbl | MR

[2]A.F. Beardon, Iteration of Rational Functions, Graduate Texts in Math. 132, Springer-Verlag, 1991. | Zbl | MR

[3]A.D. Brjuno, Analytic form of differential equations, Transaction Moscow Math. Soc., 25 (1971), 131-288. | Zbl | MR

[4]C. Camacho, On the local structure of conformal mappings and holomorphic vector fields, Astérisque, 59-60 (1978), 83-84. | Zbl | MR | mathdoc-id

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

[6]D. Cerveau, P. Sad, Problèmes de modules pour les formes différentielles singulières dans le plan complexe, Comment. Math. Helvetici, 61 (1986), 222-253. | Zbl | MR

[7]J. Écalle, Les fonctions Résurgentes I-III, preprints in Université de Paris, Orsay, 1985.

[8]P. Fatou, Sur les équations fonctionnelles, Bull. S.M.F., (1919), 161-271 48 (1920), 33-94, 208-304. | JFM | mathdoc-id

[9]Xavier Gomez-Mont, The transverse dynamics of a holomorphic flow, Ann. Math., (1988), 49-92, 127. | Zbl | MR

[10]Yu. S. Il'Yashenko, The topology of phase portraits of analytic differential equations in the complex projective plane, Trudy Sem Petrovsky, 4 (1978), 83-136. | Zbl | MR

[11]Yu.S. Il'Yashenko, The finiteness problem for limit cycles of polynomial vector fields on the plane, germs of saddle resonant vector fields and non-Hausdorff Riemann surfaces, Lecture Notes in Math. No 1060, 290-305. | Zbl | MR

[12]Yu.S. Il'Yashenko, Finiteness Theorems for Limit Cycles, Translations of Mathematical Monographs, AMS, 94, 1991. | Zbl | MR

[13]T. Kimura, On the iteration of analytic functions, Funk. Equacioj, 14-3 (1971), 197-238. | Zbl | MR

[14]V.P. Kostov, Versal deformation of differential forms of degree α on a line, Funct. Anal. App., 18 (4) (1985), 335-337. | Zbl | MR

[15]J.F. Mattei, R. Moussu, Holonomie et intégrales première, Ann. Sc. Ec. Norm. Sup., 13 (1980), 469-523. | mathdoc-id | Zbl | MR | EuDML

[16]J. Martinet, Remarques sur la bifurcation Noeud-col dans le domaine complexe, Asterisque, 150-151 (1987), 131-149. | Zbl | MR | mathdoc-id

[17]I. Nakai, On toplogical types of polynomial mappings, Topology, 23, No. 1 (1984), 45-66. | Zbl | MR

[18]I. Nakai, Topology of complex webs of codimension one and geometry of projective space curves, Topology, 26 (4) (1987), 475-504. | Zbl | MR

[19]A. Lins Neto, Construction of singular holomorphic vector fields and foliations in dimension two, J. Differential Geometry, 26 (1987), 1-31. | Zbl | MR

[20]A.A Scherbakov, Topological and analytic conjugation of non commutative groups of conformal mappings, Trudy Sem. Petrovsk, 10 (1984), 170-192, 238-239. | Zbl | MR

[21]A.A. Scherbakov, On the density of an orbit of a pseudogroup of conformal mappings and a generalization of the Hudai-Verenov theorem, Vestnik Movskovskogo Universiteta. Mathematika, 31, No.4 (1982), 10-15. | Zbl | MR

[22]S.M. Voronin, Analytic classification of germs of maps (ℂ,0) →(ℂ,0) with identical linear part, Funct. Anal., 15, No.1 (1981), 1-17. | Zbl | MR

[23]S.M. Voronin, Analytic classification of pairs of involutions and its applications, Funct. Anal., 16, No.2 (1982), 94-100. | Zbl | MR

[24]M.O. Hudai-Verenov, A property of the solution of a differential equation, Mat. Sb., 56(98) : 3 (1962), 301-308. | MR | EuDML | Zbl

Cité par Sources :