On Peixoto’s conjecture for flows on non-orientable 2-manifolds

Gutierrez, Carlos; Pires, Benito

doi:10.1090/S0002-9939-04-07687-7

On Peixoto’s conjecture for flows on non-orientable 2-manifolds
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by Carlos Gutierrez and Benito Pires PDF

Proc. Amer. Math. Soc. 133 (2005), 1063-1074 Request permission

Abstract:

Contrary to the case of vector fields on orientable compact $2$-manifolds, there is a smooth vector field $X$ on a non-orientable compact $2$-manifold with a dense orbit (and therefore without closed orbits) whose phase portrait –up to topological equivalence– remains intact under a one-parameter family of twist perturbations localized in a flow box of $X.$

References

A. Pierre, M. Malkin and E. Zhuzhoma. On the $C^r$–closing lemma for surface flows and expansions of points of the circle at infinity. Preprint IML-2001.
S. Kh. Aranson and E. V. Zhuzhoma, On the $C^r$-closing lemma and the Koebe-Morse coding of geodesics on surfaces, J. Dynam. Control Systems 7 (2001), no. 1, 15–48. MR 1817328, DOI 10.1023/A:1026693521641
Flavio Abdenur, Artur Avila, and Jairo Bochi, Robust transitivity and topological mixing for $C^1$-flows, Proc. Amer. Math. Soc. 132 (2004), no. 3, 699–705. MR 2019946, DOI 10.1090/S0002-9939-03-07187-9
C. R. Carroll, Rokhlin towers and $\scr C^r$ closing for flows on $T^2$, Ergodic Theory Dynam. Systems 12 (1992), no. 4, 683–706. MR 1200337, DOI 10.1017/S0143385700007033
John Franks, Rotation numbers and instability sets, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 3, 263–279. MR 1978565, DOI 10.1090/S0273-0979-03-00983-2
Carlos Gutiérrez, Structural stability for flows on the torus with a cross-cap, Trans. Amer. Math. Soc. 241 (1978), 311–320. MR 492303, DOI 10.1090/S0002-9947-1978-0492303-2
Carlos Gutierrez, On $C^r$-closing for flows on 2-manifolds, Nonlinearity 13 (2000), no. 6, 1883–1888. MR 1794837, DOI 10.1088/0951-7715/13/6/301
Carlos Gutiérrez, Smoothing continuous flows on two-manifolds and recurrences, Ergodic Theory Dynam. Systems 6 (1986), no. 1, 17–44. MR 837974, DOI 10.1017/S0143385700003278
C. Gutierrez, A counter-example to a $C^2$ closing lemma, Ergodic Theory Dynam. Systems 7 (1987), no. 4, 509–530. MR 922363, DOI 10.1017/S0143385700004181
Carlos Gutiérrez, Smooth nonorientable nontrivial recurrence on two-manifolds, J. Differential Equations 29 (1978), no. 3, 388–395. MR 507486, DOI 10.1016/0022-0396(78)90048-7
C. Gutierrez and B. F. Pires. On a $C^r$–Connecting Lemma for Flows on a Non-Orientable 2-Manifold. To be written.
C. Gutiérrez and J. Sotomayor, An approximation theorem for immersions with stable configurations of lines of principal curvature, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 332–368. MR 730276, DOI 10.1007/BFb0061423
C. Gutierrez and J. Sotomayor. Lines of Curvature and Umbilical Points on Surfaces. $18^{0}$ Colóquio Brasileiro de Matemática, IMPA, 1991.
Shuhei Hayashi, Connecting invariant manifolds and the solution of the $C^1$ stability and $\Omega$-stability conjectures for flows, Ann. of Math. (2) 145 (1997), no. 1, 81–137. MR 1432037, DOI 10.2307/2951824
Shuhei Hayashi, Correction to: “Connecting invariant manifolds and the solution of the $C^1$ stability and $\Omega$-stability conjectures for flows” [Ann. of Math. (2) 145 (1997), no. 1, 81–137; MR1432037 (98b:58096)], Ann. of Math. (2) 150 (1999), no. 1, 353–356. MR 1715329, DOI 10.2307/121106
Shuhei Hayashi, A $C^1$ make or break lemma, Bol. Soc. Brasil. Mat. (N.S.) 31 (2000), no. 3, 337–350. MR 1817092, DOI 10.1007/BF01241633
Michael-R. Herman, Exemples de flots hamiltoniens dont aucune perturbation en topologie $C^\infty$ n’a d’orbites périodiques sur un ouvert de surfaces d’énergies, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 13, 989–994 (French, with English summary). MR 1113091
Michael-R. Herman, Différentiabilité optimale et contre-exemples à la fermeture en topologie $C^\infty$ des orbites récurrentes de flots hamiltoniens, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 1, 49–51 (French, with English summary). MR 1115947
Ricardo Mañé, An ergodic closing lemma, Ann. of Math. (2) 116 (1982), no. 3, 503–540. MR 678479, DOI 10.2307/2007021
Ricardo Mañé, On the creation of homoclinic points, Inst. Hautes Études Sci. Publ. Math. 66 (1988), 139–159. MR 932137
Nelson G. Markley, The Poincaré-Bendixson theorem for the Klein bottle, Trans. Amer. Math. Soc. 135 (1969), 159–165. MR 234442, DOI 10.1090/S0002-9947-1969-0234442-1
Fernando Oliveira, On the generic existence of homoclinic points, Ergodic Theory Dynam. Systems 7 (1987), no. 4, 567–595. MR 922366, DOI 10.1017/S0143385700004211
Fernando Oliveira, On the $C^\infty$ genericity of homoclinic orbits, Nonlinearity 13 (2000), no. 3, 653–662. MR 1758993, DOI 10.1088/0951-7715/13/3/308
M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1 (1962), 101–120. MR 142859, DOI 10.1016/0040-9383(65)90018-2
Dennis Pixton, Planar homoclinic points, J. Differential Equations 44 (1982), no. 3, 365–382. MR 661158, DOI 10.1016/0022-0396(82)90002-X
Charles C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math. 89 (1967), 1010–1021. MR 226670, DOI 10.2307/2373414
Charles Pugh, Against the $C^{2}$ closing lemma, J. Differential Equations 17 (1975), 435–443. MR 368079, DOI 10.1016/0022-0396(75)90054-6
Charles C. Pugh, A special $C^{r}$ closing lemma, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 636–650. MR 730291, DOI 10.1007/BFb0061438
Charles C. Pugh, The $C^1$ connecting lemma, J. Dynam. Differential Equations 4 (1992), no. 4, 545–553. MR 1187222, DOI 10.1007/BF01048259
Charles C. Pugh and Clark Robinson, The $C^{1}$ closing lemma, including Hamiltonians, Ergodic Theory Dynam. Systems 3 (1983), no. 2, 261–313. MR 742228, DOI 10.1017/S0143385700001978
Christian Bonatti and Sylvain Crovisier, Recurrence and genericity, C. R. Math. Acad. Sci. Paris 336 (2003), no. 10, 839–844 (English, with English and French summaries). MR 1990025, DOI 10.1016/S1631-073X(03)00203-6
Clark Robinson, Closing stable and unstable manifolds on the two sphere, Proc. Amer. Math. Soc. 41 (1973), 299–303. MR 321141, DOI 10.1090/S0002-9939-1973-0321141-7
Floris Takens, Homoclinic points in conservative systems, Invent. Math. 18 (1972), 267–292. MR 331435, DOI 10.1007/BF01389816
Lan Wen and Zhihong Xia, $C^1$ connecting lemmas, Trans. Amer. Math. Soc. 352 (2000), no. 11, 5213–5230. MR 1694382, DOI 10.1090/S0002-9947-00-02553-8