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

Published online by Cambridge University Press:  06 August 2013

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)

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} $.

Type
Research Article
Copyright
©Cambridge University Press 2013 

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