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Separatrices for non solvable dynamics on \(\mathbb{C},0\). (English) Zbl 0804.57022

We define the separatrices for pseudogroups of diffeomorphisms of open neighbourhoods of the origin in the complex plane \(\mathbb{C}\) and prove their existence for non solvable pseudogroups (Theorem 1). This extends a result by A. A. Shcherbakov [Mosc. Univ. Math. Bull. 37, No. 4, 10- 16 (1982); translation from Vestn. Mosk. Univ., Ser. I 1982, No. 4, 10-15 (1982; Zbl 0517.30009)] accurately. Our method also applies to prove the topological rigidity theorem for generic pseudogroups attributed to Shcherbakov [Tr. Semin. Im. I. G. Petrovskogo 10, 170-196 (1984; Zbl 0568.30010)].
Reviewer: I.Nakai (Sapporo)

MSC:

57S99 Topological transformation groups
58H05 Pseudogroups and differentiable groupoids
37-XX Dynamical systems and ergodic theory
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[1] [1], Fractional iteration near a fixpoint of multiplier 1, J. Australian Math. Soc., 4 (1964), 143-148. · Zbl 0134.05402
[2] [2], Iteration of Rational Functions, Graduate Texts in Math. 132, Springer-Verlag, 1991. · Zbl 0742.30002
[3] [3], Analytic form of differential equations, Transaction Moscow Math. Soc., 25 (1971), 131-288. · Zbl 0272.34018
[4] [4], On the local structure of conformal mappings and holomorphic vector fields, Astérisque, 59-60 (1978), 83-84. · Zbl 0415.30015
[5] [5], , Groupes d’automorphismes de ℂ,0 et équations différentielles y dy +...= 0, Bull. Soc. Math. France, 116 (1988). · Zbl 0696.58011
[6] [6], , Problèmes de modules pour les formes différentielles singulières dans le plan complexe, Comment. Math. Helvetici, 61 (1986), 222-253. · Zbl 0604.58004
[7] [7], Les fonctions Résurgentes I-III, preprints in Université de Paris, Orsay, 1985.
[8] [8], Sur les équations fonctionnelles, Bull. S.M.F., (1919), 161-271 48 (1920), 33-94, 208-304. · JFM 47.0921.02
[9] [9], The transverse dynamics of a holomorphic flow, Ann. Math., (1988), 49-92, 127. · Zbl 0639.32013
[10] [9] , Algebraic solutions of differential equations (p-curvature and the Hodge filtration), Invent. Math., 18 (1972), 1-118. · Zbl 0418.34007
[11] [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. · Zbl 0588.34024
[12] [12], Finiteness Theorems for Limit Cycles, Translations of Mathematical Monographs, AMS, 94, 1991. · Zbl 0743.34036
[13] [13], On the iteration of analytic functions, Funk. Equacioj, 14-3 (1971), 197-238. · Zbl 0237.30008
[14] [14], Versal deformation of differential forms of degree α on a line, Funct. Anal. App., 18 (4) (1985), 335-337. · Zbl 0573.58002
[15] [15], , Holonomie et intégrales première, Ann. Sc. Ec. Norm. Sup., 13 (1980), 469-523. · Zbl 0458.32005
[16] [16], Remarques sur la bifurcation Noeud-col dans le domaine complexe, Asterisque, 150-151 (1987), 131-149. · Zbl 0655.58025
[17] [17], On toplogical types of polynomial mappings, Topology, 23, No. 1 (1984), 45-66. · Zbl 0531.58004
[18] [18], Topology of complex webs of codimension one and geometry of projective space curves, Topology, 26 (4) (1987), 475-504. · Zbl 0647.57018
[19] [19], Construction of singular holomorphic vector fields and foliations in dimension two, J. Differential Geometry, 26 (1987), 1-31. · Zbl 0625.57012
[20] [20], Topological and analytic conjugation of non commutative groups of conformal mappings, Trudy Sem. Petrovsk, 10 (1984), 170-192, 238-239. · Zbl 0568.30010
[21] [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. · Zbl 0517.30009
[22] [22], Analytic classification of germs of maps (ℂ,0) →(ℂ,0) with identical linear part, Funct. Anal., 15, No.1 (1981), 1-17. · Zbl 0463.30010
[23] [23], Analytic classification of pairs of involutions and its applications, Funct. Anal., 16, No.2 (1982), 94-100. · Zbl 0521.30010
[24] [24], A property of the solution of a differential equation, Mat. Sb., 56(98) : 3 (1962), 301-308. · Zbl 0111.27902
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