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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Holomorphic flows in $ {\bf C}\sp 3,0$ with resonances

Author: Júlio Cesar Canille Martins
Journal: Trans. Amer. Math. Soc. 329 (1992), 825-837
MSC: Primary 32L30; Secondary 58F18
MathSciNet review: 1073776
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Abstract: The topological classification, by conjugacy, of the germs of holomorphic diffeomorphisms $ f:\,{{\mathbf{C}}^2},0 \to {{\mathbf{C}}^2},0$ with $ df(0) = \operatorname{diag} ({\lambda _1},{\lambda _2})$, where $ {\lambda _1}$ is a root of unity and $ \vert{\lambda _2}\vert \ne 1$ is given.

This type of diffeomorphism appears as holonomies of singular foliations $ {\mathcal{F}_X}$ induced by holomorphic vector fields $ X:{{\mathbf{C}}^3},0 \to {{\mathbf{C}}^3},0$ normally hyperbolic and resonant. An explicit example of a such vector field without holomorphic invariant center manifold is presented.

We prove that there are no obstructions in the holonomies for $ {\mathcal{F}_X}$ to be topologically equivalent to a product type foliation.

References [Enhancements On Off] (What's this?)

  • [1] C. Camacho and P. Sad, Topological classification and bifurcations of holomorphic flows with resonances in $ {{\mathbf{C}}^2}$, Invent. Math. 67 (1982), 447-472. MR 664115 (84k:58177)
  • [2] C. Camacho, On the local structure of conformal mappings and holomorphic vector fields in $ {{\mathbf{C}}^2}$, Astérisque 59-60 (1978). MR 542732 (81d:58016)
  • [3] J. Martinet and J. P. Ramis, Classification analytique des équations differentielles non linéaires réssonantes du premier ordre, Ann. Sci. Ecole Norm. Sup. 16 (1983), 571-621. MR 740592 (86k:34034)
  • [4] A. D. Brjuno, Analytical form of differential equations, Trans. Moscow Math. Soc. 25 (1971), 131-288. MR 0377192 (51:13365)
  • [5] J. Palis and F. Takens, Topological equivalence of normally hyperbolic dynamical systems, Topology 16 (1977). MR 0474409 (57:14049)
  • [6] F. Dumortier, P. R. Rodrigues and R. Roussarie, Germs of diffeomorphisms in the plane, Lecture Notes in Math., vol. 902, Springer-Verlag, Berlin and New York, 1981. MR 653474 (83f:58008)
  • [7] M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Math., vol. 583, Springer-Verlag, Berlin and New York, 1977. MR 0501173 (58:18595)
  • [8] J. C. Canille Martins, Contribuição ao estudo local de fluxos holomorfos com ressonância, Tese de Doutorado, IMPA-CNPq, Brasil.

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Article copyright: © Copyright 1992 American Mathematical Society

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