On the structure of germs of vector fields in ℝ³ whose linear part generates rotations

Abstract

Our aim is to study germs of singularities of vector fields in ℝ³ whose linear part generates a 1-parameter group of rotations.

We describe how under very general conditions the ∞-jet of the vector field can give information as well on the existence of an invariant C^℞ line and invariant C^℞ cones as on the topology of the singularity. In finite codimension the weak-C°-equivalence class (which is the same as the weak-C°-conjugacy class) is revealed to be determined by a finite jet.

The same is true for the C°-equivalence class of germs in normal form.

However the genuine C°-equivalence class is not necessarily determined by a finite jet, even not by the ∞-jet. There exist non-stabilisable 9-jets, unavoidable in generic 60-parameter families of vector fields on 3-manifolds.