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The topology of holomorphic flows with singularity. (English) Zbl 0411.58018


MSC:

37C10 Dynamics induced by flows and semiflows
32S05 Local complex singularities
57R30 Foliations in differential topology; geometric theory
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References:

[1] A. D. Brjuno, Analytical form of differential equations,Trudy Moscow Math. Obšč (Trans. Moscow Math. Soc.), vol. 25 (1971), p. 131–288.
[2] C. Camacho, N. H. Kuiper, J. Palis, La topologie du feuilletage d’un champ de vecteurs holomorphe près d’une singularité,C.R. Acad. Sc. Paris, t. 282 A, p. 959–961. · Zbl 0353.32015
[3] Topological properties ofR 2-actions are studied in:C. Camacho, OnR k {\(\times\)}Z l-actions,Proceed. Symp. on Dynamical Systems, Salvador 1971, Ed. Peixoto, p. 23–70.G. Palis, Linearly induced vector fields andR 2-actions on spheres, to appear inJ. Diff. Geom.C. Camacho, Structural stability theorems for integrable differential forms on 3-manifolds, to appear inTopology.
[4] H. Dulac, Solutions d’un système d’équations différentielles dans le voisinage des valeurs singulières,Bull. Soc. Math. France,40 (1912), 324–383. · JFM 43.0391.01
[5] J. Guckenheimer, Hartman’s theorem for complex flows in the Poincaré domain,Compositio Math.,24 (1972), p. 75–82. · Zbl 0239.58007
[6] A real analogue of theorem III is the classical Grobman-Hartman theorem:P. Hartman,Proc. AMS,11 (1960), p. 610–620.
[7] Topological properties ofreal linear flows onR n are studied in:N. H. Kuiper,Manifolds Tokyo, Proceedings Int. Conference, Math. Soc. Japan (1973), p. 195–204, and:N. N. Ladis,Differentialnye Uravnenya Volg. (1973), p. 1222–1235.
[8] D. Lieberman, Holomorphic vector fields on projective varieties,Proc. Symp. Pure Math., XXX (1976), 273–276.
[9] The invariant of chapter I goes back to an invariant in the study of stability in one parameter families of diffeomorphisms:S. Newhouse, J. Palis, F. Takens, to appear. See also:J. Palis,A differentiable invariant of topological conjugacies and moduli of stability, preprint IMPA.
[10] J. Palis, S. Smale, Structural stability theorems,Global Analysis, Symp. Pure Math., AMS, vol. XIV (1970), p. 223–231. · Zbl 0214.50702
[11] H. Poincaré, Sur les propriétés des fonctions définies par les équations aux différences partielles, thèse, Paris, 1879 =OEuvres complètes, I, p. xcix–cv.
[12] H. Russmann,On the convergence of power series transformations of analytic mappings near a fized point into a normal form, Bures-sur-Yvette, preprint I.H.E.S.
[13] C. L. Siegel, Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung,Göttingen, Nachr. Akad. Wiss., Math. Phys. Kl. (1952), p. 21–30. · Zbl 0047.32901
[14] C. L. Siegel, J. Moser,Celestial mechanics (1971) (English edition of:C. L. Siegel,Vorlesungen über Himmelsmechanik, 1954, Springer Verlag).
[15] Ю. С. Ильященко ЭАМЕЧАНИЯ О ТОПОЛоГИИ ОСОЯЫХ ТОЧЕК АНАЛИТИЧЕСКИХ ДИФФЕРЕНЦИАЛьНЫХ УРАВЕНИЙ В КОМПЛЕКСНОЙ ОьЛАСТИ И ТЕОРЕМА ЛАДИСА,Фенкццональныŭ аналц§rt; ц ео прулоценця, T. 11, ВЫН 2, 1977, 28–38. · Zbl 0308.02032
[16] J. Guckenheimer, On holomorphic vector fields onCP(2),An. Acad. Brasil. Cienc.,42 (1970), p. 415–420.
[17] F. Dumortier, R. Roussarie, Smooth linearization of germs ofR 2-actions and holomorphic vector fields, to appear. · Zbl 0418.58015
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