Smooth invariant curves of singularities of vector fields on [formula omitted

Abstract

We describe singularities of C∞ vector fields, mainly in ℝ3, in the neighbourhood of a given « generalized » direction (this is: the image of a C∞ germ γ:(ℝ+,0)→(ℝ3,0) with γ′(0) ≠ 0). It is a local study. One of the major results is: if X is a germ in 0∈ℝ3 of a C∞ vector field and if X is not infinitely flat along a direction D then there exists a cone of finite contact around D in which four specific situations can occur. In three of these situations D is formally invariant under X (with formally we mean: up to the level of formal Taylor series) and there exists a C∞ one-dimensional invariant manifold having infinite contact with D. In particular, we obtain that the existence of a formally invariant direction D always implies the existence of a « real life » invariant direction having infinite contact with D, provided that X is not infinitely flat along D. Using the blowing up method for singularities of vector fields we reduce a singularity always to either a « flow box » or to a singularity with nonzero 1-jet and with a formally invariant direction. Finally we give topological models for the obtained situations.I would like to thank Freddy Dumortier for suggesting me the problem and for his valuable help.