no. 4
The Poincaré-Bendixson theorem and arational foliations on the sphere
Annales de l'Institut Fourier, Tome 46 (1996) no. 4, pp. 1159-1181.

Nous considérons les feuilletages de la 2-sphère avec un nombre fini de singularités non-orientables. Nous établissons un théorème de type Poincaré-Bendixson. En particulier, nous résolvons un problème de H. Rosenberg concernant les labyrinthes.

Foliations on the 2-sphere with a finite number of non-orientable singularities are considered. For this class a Poincaré-Bendixson theorem is established. In particular, the work gives an answer to a problem of H. Rosenberg concerning labyrinths.

@article{AIF_1996__46_4_1159_0,
     author = {Nikolaev, Igor},
     title = {The {Poincar\'e-Bendixson} theorem and arational foliations on the sphere},
     journal = {Annales de l'Institut Fourier},
     pages = {1159--1181},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {46},
     number = {4},
     year = {1996},
     doi = {10.5802/aif.1544},
     zbl = {0853.57027},
     mrnumber = {1415961},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1544/}
}
TY  - JOUR
AU  - Nikolaev, Igor
TI  - The Poincaré-Bendixson theorem and arational foliations on the sphere
JO  - Annales de l'Institut Fourier
PY  - 1996
SP  - 1159
EP  - 1181
VL  - 46
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1544/
DO  - 10.5802/aif.1544
LA  - en
ID  - AIF_1996__46_4_1159_0
ER  - 
%0 Journal Article
%A Nikolaev, Igor
%T The Poincaré-Bendixson theorem and arational foliations on the sphere
%J Annales de l'Institut Fourier
%D 1996
%P 1159-1181
%V 46
%N 4
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1544/
%R 10.5802/aif.1544
%G en
%F AIF_1996__46_4_1159_0
Nikolaev, Igor. The Poincaré-Bendixson theorem and arational foliations on the sphere. Annales de l'Institut Fourier, Tome 46 (1996) no. 4, pp. 1159-1181. doi : 10.5802/aif.1544. http://www.numdam.org/articles/10.5802/aif.1544/

[1] A.A. Andronov, L.S. Pontryagin, Systèmes grossiers, Comptes Rendus (Doklady) de l'Académie des Sciences de l'URSS, 14-5 (1937), 247-250. | JFM | Zbl

[2] S.H. Aranson, Trajectories on the non-orientable two-dimensional manifolds, Mat. Sbornik, 80 (1969), 314-333 [Russian]. | EuDML | MR | Zbl

[3] S.H. Aranson, I.U. Bronstein, I.V. Nikolaev, E.V. Zhuzhoma, Qualitative theory of foliations on surfaces, Plenum Publishing Corporation (to appear). | Zbl

[4] S.H. Aranson, V.Z. Grines, Topological classification of cascades on closed two-dimensional manifolds, Uspekhi Mat. Nauk, 45 (1990), 3-32 [Russian]. Translation in: Russian Math. Surveys. | MR | Zbl

[5] S.H. Aranson, E.V. Zhuzhoma, About structure of quasiminimal sets of foliations on surfaces, Mat. Sbornik, 185, n° 8 (1994), 31-62 [Russian]. | MR | Zbl

[6] S.H. Aranson, M.I. Malkin, E.V. Zhuzhoma, Local structure and smoothness preventing quasiminimality for flows on the torus, Differents. Uravn. : Differential Equations 29, n° 6 (1993), 789-791. | MR | Zbl

[7] V.I. Arnold, Yu. S. Ilyashenko, Ordinary Differential Equations, in : Dynamical Systems 1, Encyclopaedia of Math. Sciences, vol. 1, Springer Verlag (1988). | Zbl

[8] I.U. Bronstein, I.V. Nikolaev, Structurally stable fields of line elements on surfaces, Nonlinear Analysis (submitted). | MR | Zbl

[9] I.U. Bronstein, I.V. Nikolaev, Orbital normal forms and bifurcations of the fields of line elements in the plane, Differents. Uravn. 31, n° 6 (1995), 934-938: Differential Equations, 31, n° 6 (1995), 874-878. | MR | Zbl

[10] I.U. Bronstein, I.V. Nikolaev, Smooth orbital equivalence in the vicinity of a critical point in the plane, Differents. Uravn. 30, n° 8 (1994), 1462-1464 : Differential Equations 30, n° 8 (1994), 1357-1360. | Zbl

[11] A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures et Appl., 11, ser. 9 (1932), 333-375. | JFM | Numdam

[12] F. Dumortier, Singularities of vector fields on the plane, J. Diff. Eq., 23 (1977), 53-106. | MR | Zbl

[13] V. Guinez, Positive quadratic differential forms and foliations with singularities on surfaces, Trans. Amer. Math. Soc., 309 n° 2 (1988), 477-502. | MR | Zbl

[14] V. Guinez, Nonorientable polynomial foliations on the plane, J. Diff. Eq., 87, n° 2 (1990), 391-411. | MR | Zbl

[15] C. Gutiérrez, Structural stability for line fields with singularities on two-dimensional manifolds, Lect. Notes in Math., 468 (1975), 17-19. | Zbl

[16] C. Gutiérrez, Foliations on surfaces having exceptional leaves, Lect. Notes in Math., 1331 (1988), 73-85. | MR | Zbl

[17] C. Gutiérrez, J. Sotomayor, An approximation theorem for immersions with stable configurations of lines of principal curvature, Lect. Notes in Math., 1007, 332-368. | MR | Zbl

[18] C. Gutiérrez, J. Sotomayor, Structurally stable configurations of lines of principal curvature, Astérisque, 98-99 (1982), 195-215. | Numdam | MR | Zbl

[19] A. Hurwitz, Ueber Riemann'sche Flachen mit gegebenen Verzweigungspunkten, Math. Annalen, 39 (1891), 1-61. | JFM

[20] A.A. Kadyrov, Critical points of differential equations with unoriented trajectories in the plane, Differential Equations, 19, n° 12 (1983), 1473-1483. | MR | Zbl

[21] A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, 1995. | MR | Zbl

[22] R. Langevin, C. Possani, Courbure totale de feuilletages et enveloppes, C. R. Acad. Sci. Paris., ser. I Math., 309, n° 13 (1989), 821-824. | MR | Zbl

[23] G. Levitt, Pantalons et feuilletages des surfaces, Topology, 21 (1982), 9-33. | MR | Zbl

[24] G. Levitt, Feuilletages des surfaces, Ann. Inst. Fourier, 32-2 (1982), 179-217. | Numdam | MR | Zbl

[25] G. Levitt, La decomposition dynamique et la différentiabilité des feuilletages des surfaces, Ann. Inst. Fourier, 37-3 (1987), 85-116. | Numdam | MR | Zbl

[26] G. Levitt, H. Rosenberg, Differentiability and topology of labyrinths in the disc and annulus, Topology, 26, n° 2 (1987), 173-186. | MR | Zbl

[27] N. Markley, The Poincaré-Bendixson theorem for the Klein bottle, Trans. Amer. Math. Soc., 135 (1969), 159-165. | MR | Zbl

[28] M. Martens, S. Van Strien, W. De Melo, P. Mendes, On Cherry flows, Ergod. Th. Dynam. Sys., 10 (1990), 531-554. | MR | Zbl

[29] K.R. Meyer, Energy functions for Morse-Smale systems, Amer. J. of Math., 90 (1968), 1031-1040. | MR | Zbl

[30] I.V. Nikolaev, Foliations with singularities of semi-integer index, CRM-2206, Univ. de Montréal, preprint (1994), 33 p.

[31] I.V. Nikolaev, Qualitative Methods in the Geometric Theory of Foliations on Surfaces (in preparation).

[32] V.V. Nemytskii, V.V. Stepanov, Qualitative Theory of Differential Equations, Princeton Univ. Press, 1960. | MR | Zbl

[33] J. Palis, W. De Melo, Geometric Theory of Dynamical Systems. An Introduction, Springer Verlag, 1982. | Zbl

[34] M.M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 1 (1963), 101-120. | MR | Zbl

[35] R.V. Plykin, Sources and sinks of A-diffeomorphisms on surfaces, Mat. Sbornik, 94, n° 2 (1974), 243-264. | MR | Zbl

[36] H. Rosenberg, Labyrinths in the disc and surfaces, Annals of Math., 117, n° 1 (1983), 1-33. | MR | Zbl

[37] W.P. Thurston, On the geometry and dynamics of diffeomorphisms on surfaces, Bull. Amer. Math. Soc. 19 (1988), 417-431. | MR | Zbl

[38] J.-C. Yoccoz, Il n'y a pas de contre-exemple de Denjoy analytique, C. R. Acad. Sc. Paris, 298, ser. I, n° 7 (1984), 141-144. | MR | Zbl

Cité par Sources :