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Generic pseudogroups on $( \mathbb{C} , 0)$ and the topology of leaves

Part of:Complex dynamical systemsSmooth dynamical systems: general theoryDifferential equations in the complex domain

Published online by Cambridge University Press:  01 July 2013

J.-F. Mattei,
J. C. Rebelo and
H. Reis
J.-F. Mattei
Affiliation:
Institut de Mathématiques de Toulouse, Université Toulouse 3, 118 Route de Narbonne, F-31062 Toulouse, France email mattei@math.univ-toulouse.fr
J. C. Rebelo
Affiliation:
Institut de Mathématiques de Toulouse, Université Toulouse 3, 118 Route de Narbonne, F-31062 Toulouse, France email rebelo@math.univ-toulouse.fr
H. Reis
Affiliation:
Centro de Matemática da Universidade do Porto, Faculdade de Economia da Universidade do Porto, Portugal email hreis@fep.up.pt
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Abstract

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We show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about $0\in \mathbb{C} $ is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of consequences on the topology of leaves for a (singular) holomorphic foliation defined on a neighborhood of an invariant curve. In particular, in the classical and simplest case arising from local nilpotent foliations possessing a unique separatrix which is given by a cusp of the form $\{ {y}^{2} - {x}^{2n+ 1} = 0\} $, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves.

Keywords

pseudogroups on (ℂ,0)free groups inDiff(ℂ,0)nilpotent singularities

MSC classification

Secondary:34M35: Singularities, monodromy, local behavior of solutions, normal forms37C85: Dynamics of group actions other than ${bf Z}$ and ${bf R}$, and foliations37F75: Holomorphic foliations and vector fields

Information

Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 8 , August 2013, pp. 1401 - 1430
Copyright
© The Author(s) 2013 

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