Advertisement

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

  • Freddy DUMORTIER
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1125)

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.

Keywords

Vector Field Normal Form Conjugacy Class Characteristic Line Finite Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A.R.]
    R. Abraham, J. Robbin: Transversal mappings and flows, Benjamin, N.Y. (1967)zbMATHGoogle Scholar
  2. [B.D]
    P. Bonckaert, F. Dumortier: Smooth invariant curves for germs of vector field in IR3 whose linear part generates a rotation.Google Scholar
  3. [Du]
    F. Dumortier: Singularities of vector fields on the plane J. Diff. Eq. 23 (1977) pp. 53–106MathSciNetCrossRefzbMATHGoogle Scholar
  4. [Du2]
    F. Dumortier: Singularities of vector fields, Monografias de Matemática n° 32, IMPA, Rio de Janeiro, 1978Google Scholar
  5. [Du3]
    F. Dumortier: Non-stabilisable jets of diffeomorphisms in IR2 and of vector fields in IR3, to appearGoogle Scholar
  6. [D.R.]
    F. Dumortier, R. Roussarie: Germes de difféomorphismes et de champs de vecteurs en classe de différentiabilité finie. Annales de l'institut Fourier. Tome XXXIII, 1, 1983, p 195–267MathSciNetzbMATHGoogle Scholar
  7. [D.R.R.]
    F. Dumortier, P.R. Rodriguez, R. Roussarie Germs of diffeomorphisms in the Plane, Lecture Notes in Mathematics 902, 1981, p 1–197 Springer-VerlagMathSciNetCrossRefzbMATHGoogle Scholar
  8. [b]
    S. bojasiewics: Ensembles Semi-analytiques, IHES Lecture Notes, 1965Google Scholar
  9. [S]
    A. Seidenberg: A new decision method for elementary algebra, Ann. of Math 60 (1954) 365–374MathSciNetCrossRefzbMATHGoogle Scholar
  10. [Ta]
    F. Takens: Singularities of vector fields, Publ. Math. IHES 43 (1974) pp 47–100MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Freddy DUMORTIER
    • 1
  1. 1.Limburgs Universitair CentrumUniversitaire CampusDiepenbeekBelgium

Personalised recommendations