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Cyclic stabilizers and infinitely many hyperbolic orbits for pseudogroups on $( \mathbb{C} , 0)$

Part of: Smooth dynamical systems: general theory Differential equations in the complex domain Complex dynamical systems

Published online by Cambridge University Press:  06 August 2013

Julio C. Rebelo  and
Helena Reis
Julio C. Rebelo
Affiliation:
Institut de Mathématiques de Toulouse, 118 Route de Narbonne, F-31062 Toulouse, France (rebelo@math.univ-toulouse.fr)
Helena Reis
Affiliation:
Centro de Matemática da Universidade do Porto, Faculdade de Economia da Universidade do Porto, Portugal (hreis@fep.up.pt)
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Abstract

Consider a pseudogroup on $( \mathbb{C} , 0)$ generated by two local diffeomorphisms having analytic conjugacy classes a priori fixed in $\mathrm{Diff} \hspace{0.167em} ( \mathbb{C} , 0)$. We show that a generic pseudogroup as above is such that every point has a (possibly trivial) cyclic stabilizer. It also follows that these generic groups possess infinitely many hyperbolic orbits. This result possesses several applications to the topology of leaves of foliations, and we shall explicitly describe the case of nilpotent foliations associated to Arnold’s singularities of type ${A}^{2n+ 1} $.

Keywords

pseudogroups on (ℂ,0)fixed pointscyclic stabilizers

MSC classification

Secondary: 37C85: Dynamics of group actions other than ${bf Z}$ and ${bf R}$, and foliations 37F75: Holomorphic foliations and vector fields 34M35: Singularities, monodromy, local behavior of solutions, normal forms
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 13 , Issue 2 , April 2014 , pp. 413 - 446
Copyright
©Cambridge University Press 2013 

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