Orbit Structure of certain ${\mathbb{R}}^{2}$-actions on solid torus

Maquera, C.; Martins, L. F.

doi:10.5802/afst.1195

Geodesic

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Orbit Structure of certain

ℝ^{2}

-actions on solid torus

Maquera, C.¹ ; Martins, L. F.²

¹ Departamento de Matemática, USP – Universidade de São Paulo, Instituto de Ciências Matemáticas e de Computação, Caixa Postal 668, CEP 13560-970, São Carlos, SP, Brazil. Work supported by FAPESP of Brazil Grant 02/09425-0.
² Departamento de Matemática, UNESP – Universidade Estadual Paulista, Instituto de Biologia, Letras e Ciências Exatas, R. Cristóvão Colombo, 2265, Jardim Nazareth, 15054-000, São José do Rio Preto, SP, Brazil. Work supported by FAPESP of Brazil Grant 98/13400-5

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 3, pp. 613-633.

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Abstract (VO)
Résumé

In this paper we describe the orbit structure of $C^{2}$ -actions of $ℝ^{2}$ on the solid torus $S^{1} \times D^{2}$ having $S^{1} \times {0}$ and $S^{1} \times \partial D^{2}$ as the only compact orbits, and $S^{1} \times {0}$ as singular set.

Nous décrivons la structure des orbites des actions de class $C^{2}$ de $ℝ^{2}$ sur le tore solide $S^{1} \times D^{2}$ ayant uniquement $S^{1} \times {0}$ et $S^{1} \times \partial D^{2}$ comme orbites compacts, et $S^{1} \times {0}$ comme ensemble singulier.

MR Zbl

DOI : 10.5802/afst.1195

Affiliations des auteurs :

Maquera, C. ¹ ; Martins, L. F. ²

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     title = {Orbit {Structure} of certain ${\mathbb{R}}^{2}$-actions on solid torus},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {613--633},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
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Maquera, C.; Martins, L. F. Orbit Structure of certain ${\mathbb{R}}^{2}$-actions on solid torus. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 3, pp. 613-633. doi : 10.5802/afst.1195. https://geodesic-test.mathdoc.fr/articles/10.5802/afst.1195/

Bibliographie
Cité par

[1] Arraut (J. L.), Craizer (M.).— Foliations of $M^{3}$ defined by $ℝ^{2}$ -actions. Ann. Inst. Fourier, Grenoble, 45, 4, p. 1091-1118 (1995). | Zbl | MR | mathdoc-id

[2] Arraut (J. L.), Maquera (C. A.).— On the orbit structure of $ℝ^{n}$ -actions on $n$ -manifolds. Qual. Theory Dyn. Syst., 4, no. 2, p. 169-180 (2003). | Zbl | MR

[3] Camacho (C.), Lins Neto (A.).— Geometric Theory of Foliations. Birkhäuser, Boston, Massachusetts, 1985. | Zbl | MR

[4] Camacho (C.), Scárdua (B. A.).— On codimension one foliations with Morse singularities on three-manifolds. Topology Appl., 154, p. 1032-1040 (2007). | Zbl | MR

[5] Cearveau (D.).— Distributions involutives singulières. Ann. Inst. Fourier, 29, no.3, 261-294 (1979). | Zbl | MR | mathdoc-id

[6] Chatelet (G.), Rosenberg (H.), Weil (D.).— A classification of the topological types of $ℝ^{2}$ -actions on closed orientable $3$ -manifolds. Publ. Math. IHES, 43, p. 261-272 (1974). | Zbl | MR | mathdoc-id

[7] Haefliger (A.).— Variétés feuilletées. Ann. Scuola Norm. Sup. Pisa, Série 3, 16, p. 367-397 (1962). | Zbl | MR | mathdoc-id

[8] Katok (A.), Hasselblatt (B.).— Introduction to the modern theory of dynamical systems. Cambridge University Press, 1995. | Zbl | MR

[9] de Medeiros (A. S.).— Singular foliations, differential p-forms. Ann. Fac. Sci. Toulouse Math., (6) 9, no.3, p. 451-466 (2000). | Zbl | MR | mathdoc-id

[10] Rosenberg (H.), Roussarie (R.), Weil (D.).— A classification of closed orientable manifolds of rank two. Ann. of Math., 91, p. 449-464 (1970). | Zbl | MR

[11] Sacksteder (R.).— Foliations, pseudogroups. Amer. J. Math., 87, p. 79-102 (1965). | Zbl | MR

[12] Scardua (B.), Seade (J.).— Codimension one foliations with Bott-Morse singularities I. Pre-print (2007). | Zbl

[13] Stefan (P.).— Accessible sets, orbits,, foliations with singularities. Proc. London Math. Soc., 29, p. 699-713 (1974). | Zbl | MR

[14] Sussmann (H. J.).— Orbits of families of vector fields, integrability of distributions. Trans. Amer. Math. Soc., 180, p. 171-188 (1973). | Zbl | MR

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