Local structural stability of $C^2$ integrable 1-forms

Neto, Alcides Lins

Local structural stability of

C^{2}

integrable 1-forms

Neto, Alcides Lins

Annales de l'Institut Fourier, Volume 27 (1977) no. 2, p. 197-225

Abstract
Résumé

In this work we consider a class of germs of singularities of integrable 1-forms in $R^{n}$ which are structurally stable in class $C^{r}$ ( $r \geq 2$ if $n = 3$ , $r \geq 4$ if $n \geq 4$ ), whose 1-jet is zero at the singularity. In this class the stability depends essentially on the fact that the perturbations allowed are integrable.

Dans ce travail, on considère une classe de germes de singularités de 1-formes intégrables dans $R^{n}$ qui sont $C^{r}$ structuralement stables ( $r \geq 2$ si $n = 3$ , $r \geq 4$ si $n \geq 4$ ). Dans cette classe la stabilité dépend essentiellement de ce que les perturbations permises sont intégrables.

MR 58 #2848 | Zbl 0356.58008 | 2 citations in Numdam

DOI : https://doi.org/10.5802/aif.657

@article{AIF_1977__27_2_197_0,
     author = {Neto, Alcides Lins},
     title = {Local structural stability of $C^2$ integrable 1-forms},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {27},
     number = {2},
     year = {1977},
     pages = {197-225},
     doi = {10.5802/aif.657},
     zbl = {0356.58008},
     mrnumber = {58 \#2848},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1977__27_2_197_0}
}

Neto, Alcides Lins. Local structural stability of $C^2$ integrable 1-forms. Annales de l'Institut Fourier, Volume 27 (1977) no. 2, pp. 197-225. doi : 10.5802/aif.657. http://www.numdam.org/item/AIF_1977__27_2_197_0/

References

[1] G. Reeb, Propriétés Topologiques des Variétés Feuilletées, Actualités Sci. Ind., 1183 (1952). | MR 14,1113a | Zbl 0049.12602

[2] I. Kupka, The Singularities of Integrable Structurally Stable Pfaffian Forms, Proc. of the Nat. Acad. of Sc., vol. 52 (1964), 1431. | MR 30 #3427 | Zbl 0137.41404

[3] A. S. Medeiros, Structural Stability of Integrable Differential 1-Forms, Thesis IMPA (1974), to appear. | Zbl 0363.58007

[4] C. Camacho, On Rk ˟ Zl-Actions, Proceedings of the Salvador Symposium on Dynamical Systems (1971). | Zbl 0274.58006

[5] J. Palis, On Morse-Smale Dynamical Systems, Topology, (1969). | Zbl 0189.23902

[6] M. C. Peixoto and M. Peixoto, Structural Stability in the Plane with Enlarged Boundary Conditions, Ann. Acad. Bras. Sci., vol. 81 (1959), 135-160. | MR 21 #5794 | Zbl 0107.07102

[7] J. Sotomayor, Generic One Parameter Families of Vector Fields on Two-Dimensional Manifolds, Publ. Math. 43, IHESc. | Numdam | Zbl 0279.58008

[8] M. W. Hirsch and C. C. Pugh, Stable Manifolds and Hyperbolic Sets, Global Analysis, Proc. Symp. in Pure Math., vol. XIV, AMS (1970). | MR 42 #6872 | Zbl 0215.53001

[9] P. Hartman, Ordinary Differential Equations, edited by John Wiley and Sons Inc., 1964. | MR 30 #1270 | Zbl 0125.32102

[10] C. Camacho, Structural Stability of integrable forms on 3-manifolds, to appear. | Zbl 0391.58013