Smooth linearization of germs of R 2 -actions and holomorphic vector fields

F. Dumortier; Robert Roussarie

Annales de l'institut Fourier (1980)

  • Volume: 30, Issue: 1, page 31-64
  • ISSN: 0373-0956

Abstract

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The paper contains a generic condition permitting the linearization in class 𝒞 k , 0 ≤ k ≤ ∞ , of germs of singular infinitesimal R 2 -actions on R n ( n ≥ 2 ) and of singular holomorphic vector fields on C n ( n ≥ 1 ) . It generalizes a similar result of S. Sternberg for germs of singular (real) vector fields on R n .

How to cite

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Dumortier, F., and Roussarie, Robert. "Smooth linearization of germs of $R^2$-actions and holomorphic vector fields." Annales de l'institut Fourier 30.1 (1980): 31-64. <http://eudml.org/doc/74442>.

@article{Dumortier1980,
abstract = {The paper contains a generic condition permitting the linearization in class $\{\cal C\}^k$, $0\le k\le \infty $, of germs of singular infinitesimal $\{\bf R\}^2$-actions on $\{\bf R\}^n$$(n\ge 2)$ and of singular holomorphic vector fields on $\{\bf C\}^n$$(n\ge 1)$. It generalizes a similar result of S. Sternberg for germs of singular (real) vector fields on $\{\bf R\}^n$.},
author = {Dumortier, F., Roussarie, Robert},
journal = {Annales de l'institut Fourier},
keywords = {smooth linearization of germs of R2-actions; holomorphic vector fields},
language = {eng},
number = {1},
pages = {31-64},
publisher = {Association des Annales de l'Institut Fourier},
title = {Smooth linearization of germs of $R^2$-actions and holomorphic vector fields},
url = {http://eudml.org/doc/74442},
volume = {30},
year = {1980},
}

TY - JOUR
AU - Dumortier, F.
AU - Roussarie, Robert
TI - Smooth linearization of germs of $R^2$-actions and holomorphic vector fields
JO - Annales de l'institut Fourier
PY - 1980
PB - Association des Annales de l'Institut Fourier
VL - 30
IS - 1
SP - 31
EP - 64
AB - The paper contains a generic condition permitting the linearization in class ${\cal C}^k$, $0\le k\le \infty $, of germs of singular infinitesimal ${\bf R}^2$-actions on ${\bf R}^n$$(n\ge 2)$ and of singular holomorphic vector fields on ${\bf C}^n$$(n\ge 1)$. It generalizes a similar result of S. Sternberg for germs of singular (real) vector fields on ${\bf R}^n$.
LA - eng
KW - smooth linearization of germs of R2-actions; holomorphic vector fields
UR - http://eudml.org/doc/74442
ER -

References

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  1. [1] A.D. BRJUNO, The analytic form of differential equations, Trans. Moscow Math. Soc., Vol. 25 (1971), 131-288. Zbl0272.34018MR51 #13365
  2. [2] C. CAMACHO, On Rk x Zl-actions, Dynamical Systems, Proc. of the Salvador Symposium, Ed. M.M. Peixoto, (1973), 23-70. Zbl0274.58006
  3. [3] C. CAMACHO, N.H. KUIPER, J. PALIS, La topologie du feuilletage d'un champ de vecteurs holomorphes près d'une singularité, C.R. Acad. Sc. Paris, t. 282 (1976), 959-961. Zbl0353.32015MR54 #1301
  4. [4] C. CAMACHO, N.H. KUIPER, J. PALIS, The topology of holomorphic flows with singularity, Preprint of the I.H.E.S., Bures-sur-Yvette. Zbl0411.58018
  5. [5] F. DUMORTIER, R. ROUSSARIE, Linéarisation différentiable des germes singuliers d'actions de R2 et de champs de vecteurs holomorphes, C.R. Acad. Sc. Paris, t. 285 Série A (1977), 841. Zbl0367.58003MR58 #13136
  6. [6] J. GUCKENHEIMER, Hartman's theorem for complex flows in the Poincaré domain, Compositio Math., 24 (1972), 75-82. Zbl0239.58007MR46 #920
  7. [7] P. HARTMAN, On the local linearization of differential equations, Proc. Amer. Math. Soc., 14 (1963), 568-573. Zbl0115.29801MR27 #2693
  8. [8] M. HIRSCH, C.C. PUGH, M. SHUB, Invariant manifolds, Lecture Notes in Math., 583 (1977). Zbl0355.58009MR58 #18595
  9. [9] N. KOPELL, Commuting diffeomorphisms global analysis, Proc. of Symp. in Pure Math., XIV, (1970), 165-184. Zbl0225.57020MR42 #5285
  10. [10] R. ROUSSARIE, Modèles locaux de champs et de formes, Astérisque, Vol. 30 (1975). Zbl0327.57017MR55 #13444
  11. [11] R. ROUSSARIE, Singularités des germes de champs de vecteurs, Bol. Soc. Brasileira de Matemática (to appear). Zbl0414.58025
  12. [12] G.L. SIEGEL, Nachr. Akad. Wiss Göttingen, Math. Phys., I (1952), 21-30. Zbl0047.32901
  13. [13] S. SMALE, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967). Zbl0202.55202
  14. [14] S. STERNBERG, On the structure of local homeomorphisms of Euclidean n-space II, J. of Math., Vol. 80 (1958), 623-631. Zbl0083.31406MR20 #3336

Citations in EuDML Documents

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  1. Didier Arnal, Mabrouk Benammar, Mohamed Selmi, Normalisation d'une représentation non linéaire d'une algèbre de Lie
  2. Aubin Arroyo, Adolfo Guillot, A rigidity phenomenon for germs of actions of R 2
  3. Marc Chaperon, C k -conjugacy of holomorphic flows near a singularity
  4. Masafumi Yoshino, Todor Gramchev, Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks
  5. Laurent Stolovitch, Singular complete integrability

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