On the structure of germs of vector fields in ℝ3 whose linear part generates rotations
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.
KeywordsVector Field Normal Form Conjugacy Class Characteristic Line Finite Type
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