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Transactions of the American Mathematical Society

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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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1073776-0
Article copyright: © Copyright 1992 American Mathematical Society