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\(\omega\)-limit sets from nonrecurrent points of flows on manifolds. (English) Zbl 1085.37013

Summary: We give a topological characterization of \(\omega\)-limit sets from nonrecurrent points of flows on manifolds. This characterization is an extension of the one obtained for surfaces by V. Jiménez López and the author [Topology Appl. 137, 187–194 (2004; Zbl 1040.37019)]. However the result is not stated in the same terms.
For the case of the \(m\)-dimensional sphere, we already gave a topological description of \(\omega\)-limit sets of nonrecurrent points in [V. Jiménez López and the author, Int. J. Bifurcation Chaos Appl. Sci. Eng. 13, 1727–1732 (2003; Zbl 1056.37007)]. This description generalized the Vinograd theorem, but it was only proved for the standard differential structure of \(\mathbb S^m\).
In this note, we obtain the same characterization for all differentiable structures as an easy consequence of the main result.

MSC:

37C10 Dynamics induced by flows and semiflows
37E35 Flows on surfaces
54H20 Topological dynamics (MSC2010)
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
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References:

[1] Anosov, D. V., Flows on closed surfaces and behavior of trajectories lifted to the universal covering plane, J. Dynam. Control Systems, 1, 125-138 (1995) · Zbl 0995.37014
[2] Bendixson, I., Sur les courbes définies par des Équations différentielles, Acta Math., 24, 1-88 (1901) · JFM 31.0328.03
[3] Beniere, J. C.; Meigniez, G., Flows without minimal set, Ergodic Theory Dynamical Systems, 19, 21-33 (1997) · Zbl 0920.58038
[4] Ding, C., The \(ω\)-limit set and the limit set of subsets, Chaos Solitons Fractals, 15, 659-661 (2003) · Zbl 1070.37008
[5] Inaba, T., An example of a flow on a non compact surface without minimal set, Ergodic Theory Dynamical Systems, 19, 31-33 (1999) · Zbl 0919.58054
[6] Jiménez López, V.; Soler López, G., Accumulation points of nonrecurrent orbits of surface flows, Topology Appl., 137, 187-194 (2004) · Zbl 1040.37019
[7] Jiménez López, V.; Soler López, G., Transitive flows on manifolds, Rev. Mat. Iberoamericana, 20, 107-130 (2004) · Zbl 1063.54032
[8] Jiménez López, V.; Soler López, G., A characterization of \(ω\)-limit sets of non-recurrent orbits in \(S^n\), Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13, 1727-1732 (2001) · Zbl 1056.37007
[9] Jiménez López, V.; Soler López, G., A topological characterization of \(ω\)-limit sets for continuous flows on the projective plane, Discrete Contin. Dynam. Systems, Added Volume, 254-258 (2001) · Zbl 1301.37028
[10] Morales, C. A.; Pacifico, M. J., Lyapunov stability of \(ω\)-limit sets, Discrete Contin. Dyn. Syst., 8, 671-674 (2002) · Zbl 1162.37302
[11] Nikolaev, I.; Zhuzhoma, E., Flows on 2-Dimensional Manifolds, Lecture Notes in Math., vol. 1705 (1999), Springer: Springer Berlin · Zbl 1022.37027
[12] Schwartz, A. J., A generalization of a Poincaré-Bendixson Theorem to closed two-dimensional manifolds, Amer. J. Math., 85, 453-458 (1963) · Zbl 0116.06803
[13] Soler López, G., Accumulation points of flows on the Klein bottle, Discrete Contin. Dynam. Systems, 9, 497-503 (2003) · Zbl 1029.37025
[14] Smith, R. A.; Thomas, S., Some examples of transitive smooth flows on differentiable manifolds, J. London Math. Soc., 37, 552-568 (1988) · Zbl 0634.58025
[15] Smith, R. A.; Thomas, S., Transitive flows on two-dimensional manifolds, J. London Math. Soc., 37, 569-576 (1988) · Zbl 0634.58026
[16] Vinograd, R. E., On the limiting behavior of an unbounded integral curve, Moskov. Gos. Univ. Uč. Zap. 155, Mat., 5, 94-136 (1952), (in Russian)
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