Holomorphic dynamics near germs of singular curves

Details Holomorphic dynamics near germs of singular curves Holomorphic dynamics near germs of singular curves

2005-02-02

arxiv.org/abs/math/0502044v1

Mathematics

Complex Variables

14 pages, 1 figure

Scientific paper

Let $M$ be a two dimensional complex manifold, $p \in M $ and \Fl a germ of holomorphic foliation of \M at $p$. Let $S\subset M$ be a germ of an irreducible, possibly singular, curve at $p$ in $M$ which is a separatrix for \Fl. We prove that if the Camacho-Sad-Suwa index $\id(\F,S,p)\not \in \Q^+\cup \{0\} $ then there exists another separatrix for \Fl at $p$. A similar result is proved for the existence of parabolic curves for germs of holomorphic diffeomorphisms near a curve of fixed points.

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