The Poincaré-Bendixson theorem and arational foliations on the sphere

 Access to full text

 Full (PDF)

1. [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. Zbl0016.11301JFM63.0728.01
2. [2] S.H. ARANSON, Trajectories on the non-orientable two-dimensional manifolds, Mat. Sbornik, 80 (1969), 314-333 [Russian]. Zbl0207.54501MR41 #3926
3. [3] S.H. ARANSON, I.U. BRONSTEIN, I.V. NIKOLAEV, E.V. ZHUZHOMA, Qualitative theory of foliations on surfaces, Plenum Publishing Corporation (to appear). Zbl0948.53048
4. [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. Zbl0705.58038MR91d:58195
5. [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]. Zbl0842.57024MR95i:57027
6. [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. Zbl0838.58034MR94k:58124
7. [7] V.I. ARNOLD, Yu. S. ILYASHENKO, Ordinary Differential Equations, in : Dynamical Systems 1, Encyclopaedia of Math. Sciences, vol. 1, Springer Verlag (1988). Zbl0718.34070
8. [8] I.U. BRONSTEIN, I.V. NIKOLAEV, Structurally stable fields of line elements on surfaces, Nonlinear Analysis (submitted). Zbl0881.57027
9. [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. Zbl0861.34019
10. [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. Zbl0852.34026
11. [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. Zbl58.1124.04JFM58.1124.04
12. [12] F. DUMORTIER, Singularities of vector fields on the plane, J. Diff. Eq., 23 (1977), 53-106. Zbl0346.58002MR58 #31276
13. [13] V. GUINEZ, Positive quadratic differential forms and foliations with singularities on surfaces, Trans. Amer. Math. Soc., 309 n° 2 (1988), 477-502. Zbl0707.57014MR89h:57021
14. [14] V. GUINEZ, Nonorientable polynomial foliations on the plane, J. Diff. Eq., 87, n° 2 (1990), 391-411. Zbl0722.57011MR91j:58127
15. [15] C. GUTIÉRREZ, Structural stability for line fields with singularities on two-dimensional manifolds, Lect. Notes in Math., 468 (1975), 17-19. Zbl0322.58008
16. [16] C. GUTIÉRREZ, Foliations on surfaces having exceptional leaves, Lect. Notes in Math., 1331 (1988), 73-85. Zbl0658.57017MR89j:57021
17. [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. Zbl0528.53002MR85b:53002
18. [18] C. GUTIÉRREZ, J. SOTOMAYOR, Structurally stable configurations of lines of principal curvature, Astérisque, 98-99 (1982), 195-215. Zbl0521.53003MR85h:58006
19. [19] A. HURWITZ, Ueber Riemann'sche Flachen mit gegebenen Verzweigungspunkten, Math. Annalen, 39 (1891), 1-61. JFM23.0429.01
20. [20] A.A. KADYROV, Critical points of differential equations with unoriented trajectories in the plane, Differential Equations, 19, n° 12 (1983), 1473-1483. Zbl0553.34019MR85h:34036
21. [21] A. KATOK, B. HASSELBLATT, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, 1995. Zbl0878.58020MR96c:58055
22. [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. Zbl0696.57013MR91h:53036
23. [23] G. LEVITT, Pantalons et feuilletages des surfaces, Topology, 21 (1982), 9-33. Zbl0473.57014MR83f:57017
24. [24] G. LEVITT, Feuilletages des surfaces, Ann. Inst. Fourier, 32-2 (1982), 179-217. Zbl0454.57015MR84g:57021
25. [25] G. LEVITT, La decomposition dynamique et la différentiabilité des feuilletages des surfaces, Ann. Inst. Fourier, 37-3 (1987), 85-116. Zbl0596.57019MR88m:57035
26. [26] G. LEVITT, H. ROSENBERG, Differentiability and topology of labyrinths in the disc and annulus, Topology, 26, n° 2 (1987), 173-186. Zbl0621.57013MR89e:57025
27. [27] N. MARKLEY, The Poincaré-Bendixson theorem for the Klein bottle, Trans. Amer. Math. Soc., 135 (1969), 159-165. Zbl0175.50101MR38 #2759
28. [28] M. MARTENS, S. VAN STRIEN, W. DE MELO, P. MENDES, On Cherry flows, Ergod. Th. Dynam. Sys., 10 (1990), 531-554. Zbl0694.58025MR91m:58134
29. [29] K.R. MEYER, Energy functions for Morse-Smale systems, Amer. J. of Math., 90 (1968), 1031-1040. Zbl0219.58004MR39 #577
30. [30] I.V. NIKOLAEV, Foliations with singularities of semi-integer index, CRM-2206, Univ. de Montréal, preprint (1994), 33 p. 
31. [31] I.V. NIKOLAEV, Qualitative Methods in the Geometric Theory of Foliations on Surfaces (in preparation). 
32. [32] V.V. NEMYTSKII, V.V. STEPANOV, Qualitative Theory of Differential Equations, Princeton Univ. Press, 1960. Zbl0089.29502MR22 #12258
33. [33] J. PALIS, W. DE MELO, Geometric Theory of Dynamical Systems. An Introduction, Springer Verlag, 1982. Zbl0491.58001
34. [34] M.M. PEIXOTO, Structural stability on two-dimensional manifolds, Topology, 1 (1963), 101-120. Zbl0107.07103MR26 #426
35. [35] R.V. PLYKIN, Sources and sinks of A-diffeomorphisms on surfaces, Mat. Sbornik, 94, n° 2 (1974), 243-264. Zbl0324.58013MR50 #8608
36. [36] H. ROSENBERG, Labyrinths in the disc and surfaces, Annals of Math., 117, n° 1 (1983), 1-33. Zbl0522.57028MR84h:57016
37. [37] W.P. THURSTON, On the geometry and dynamics of diffeomorphisms on surfaces, Bull. Amer. Math. Soc. 19 (1988), 417-431. Zbl0674.57008MR89k:57023
38. [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. Zbl0573.58023MR85j:58134