We define the separatrices for pseudogroups of diffeomorphisms of open neighbourhoods of the origin in the complex plane 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 , 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://www.numdam.org/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://www.numdam.org/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://www.numdam.org/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, Volume 44 (1994) no. 2, pp. 569-599. doi : 10.5802/aif.1410. http://www.numdam.org/articles/10.5802/aif.1410/
[1]Fractional iteration near a fixpoint of multiplier 1, J. Australian Math. Soc., 4 (1964), 143-148. | MR | Zbl
,[2]Iteration of Rational Functions, Graduate Texts in Math. 132, Springer-Verlag, 1991. | MR | Zbl
,[3]Analytic form of differential equations, Transaction Moscow Math. Soc., 25 (1971), 131-288. | MR | Zbl
,[4]On the local structure of conformal mappings and holomorphic vector fields, Astérisque, 59-60 (1978), 83-84. | Numdam | MR | Zbl
,[5]Groupes d'automorphismes de ℂ,0 et équations différentielles y dy +...= 0, Bull. Soc. Math. France, 116 (1988). | Numdam | MR | Zbl
, ,[6]Problèmes de modules pour les formes différentielles singulières dans le plan complexe, Comment. Math. Helvetici, 61 (1986), 222-253. | MR | Zbl
, ,[7]Les fonctions Résurgentes I-III, preprints in Université de Paris, Orsay, 1985.
,[8]Sur les équations fonctionnelles, Bull. S.M.F., (1919), 161-271 48 (1920), 33-94, 208-304. | JFM | Numdam
,[9]The transverse dynamics of a holomorphic flow, Ann. Math., (1988), 49-92, 127. | MR | Zbl
,[10]The topology of phase portraits of analytic differential equations in the complex projective plane, Trudy Sem Petrovsky, 4 (1978), 83-136. | MR | Zbl
,[11]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. | MR | Zbl
,[12]Finiteness Theorems for Limit Cycles, Translations of Mathematical Monographs, AMS, 94, 1991. | MR | Zbl
,[13]On the iteration of analytic functions, Funk. Equacioj, 14-3 (1971), 197-238. | MR | Zbl
,[14]Versal deformation of differential forms of degree α on a line, Funct. Anal. App., 18 (4) (1985), 335-337. | MR | Zbl
,[15]Holonomie et intégrales première, Ann. Sc. Ec. Norm. Sup., 13 (1980), 469-523. | EuDML | Numdam | MR | Zbl
, ,[16]Remarques sur la bifurcation Noeud-col dans le domaine complexe, Asterisque, 150-151 (1987), 131-149. | Numdam | MR | Zbl
,[17]On toplogical types of polynomial mappings, Topology, 23, No. 1 (1984), 45-66. | MR | Zbl
,[18]Topology of complex webs of codimension one and geometry of projective space curves, Topology, 26 (4) (1987), 475-504. | MR | Zbl
,[19]Construction of singular holomorphic vector fields and foliations in dimension two, J. Differential Geometry, 26 (1987), 1-31. | MR | Zbl
,[20]Topological and analytic conjugation of non commutative groups of conformal mappings, Trudy Sem. Petrovsk, 10 (1984), 170-192, 238-239. | MR | Zbl
,[21]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. | MR | Zbl
,[22]Analytic classification of germs of maps (ℂ,0) →(ℂ,0) with identical linear part, Funct. Anal., 15, No.1 (1981), 1-17. | MR | Zbl
,[23]Analytic classification of pairs of involutions and its applications, Funct. Anal., 16, No.2 (1982), 94-100. | MR | Zbl
,[24]A property of the solution of a differential equation, Mat. Sb., 56(98) : 3 (1962), 301-308. | EuDML | MR | Zbl
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