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Empty interior recurrence for continuous flows on surfaces. (English) Zbl 1202.37065

Summary: We characterize topologically the empty interior subsets of a compact surface S which can be \(\omega \)-limit sets of recurrent orbits (but not of nonrecurrent ones) of continuous flows on S. This culminates the classification of \(\omega \)-limit sets for surface flows.
We also show that this type of \(\omega \)-limit sets can always be realized (up to topological equivalence) by smooth flows but cannot be realized by analytic flows.

MSC:

37E35 Flows on surfaces
37B35 Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
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[1] DOI: 10.1007/BF02254658 · Zbl 0995.37014 · doi:10.1007/BF02254658
[2] DOI: 10.1070/SM1969v009n03ABEH001130 · Zbl 0207.54501 · doi:10.1070/SM1969v009n03ABEH001130
[3] Aranson S. Kh., Introduction to the Qualitative Theory of Dynamical Systems on Surfaces (1996) · Zbl 0853.58090
[4] DOI: 10.1007/BF02433487 · Zbl 0939.37024 · doi:10.1007/BF02433487
[5] DOI: 10.1006/jdeq.1997.3401 · Zbl 0913.34039 · doi:10.1006/jdeq.1997.3401
[6] Bebutov M. V., Bull. Math. Univ. Moscou 2 pp 1–
[7] DOI: 10.1007/BF02403068 · JFM 31.0328.03 · doi:10.1007/BF02403068
[8] Bohr H., Danske Vid. Selsk 14 pp 1–
[9] DOI: 10.2307/2371361 · Zbl 0017.35101 · doi:10.2307/2371361
[10] Cherry T. M., Proc. London Math. Soc. 44 pp 175–
[11] Denjoy A., J. Math. Pures Appl. 11 pp 333–
[12] Gutierrez C., Trans. Amer. Math. Soc. 24 pp 311–
[13] Gutierrez C., Erg. Th. Dyn. Syst. 6 pp 17–
[14] DOI: 10.2307/1968626 · Zbl 0015.32304 · doi:10.2307/1968626
[15] Jiménez López V., Discr. Contin. Dyn. Syst. pp 254–
[16] DOI: 10.1016/S0166-8641(03)00208-6 · Zbl 1040.37019 · doi:10.1016/S0166-8641(03)00208-6
[17] Jiménez López V., Rev. Mat. Iberoamericana 20 pp 107–
[18] Jiménez López V., Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 9 pp 515–
[19] DOI: 10.1016/j.aim.2007.06.007 · Zbl 1134.37016 · doi:10.1016/j.aim.2007.06.007
[20] Maĭer A. G., Mat. Sb. 12 pp 71–
[21] Markley N. G., Trans. Amer. Math. Soc. 135 pp 159–
[22] Marzougui H., C. R. Acad. Sci. Paris Sér. I Math. 323 pp 185–
[23] Nikolaev I., Lecture Notes in Mathematics 1705, in: Flows on Two Dimensional Manifolds, An Overview (1999) · Zbl 1022.37027 · doi:10.1007/BFb0093599
[24] DOI: 10.1007/978-3-662-04524-4 · doi:10.1007/978-3-662-04524-4
[25] Poincaré H., J. Math. Pures Appl. 2 pp 151–
[26] DOI: 10.2307/2373135 · Zbl 0116.06803 · doi:10.2307/2373135
[27] DOI: 10.3934/dcds.2003.9.497 · Zbl 1029.37025 · doi:10.3934/dcds.2003.9.497
[28] Smith R. A., J. London Math. Soc. 37 pp 569–
[29] Solncev Iu. K., Izv. Akad. Nauk SSSR 9 pp 233–
[30] DOI: 10.1016/0022-0396(70)90092-6 · Zbl 0197.49802 · doi:10.1016/0022-0396(70)90092-6
[31] Vinograd R. E., Moskov. Gos. Univ. Uc. Zap. 155, Mat. 5 pp 94–
[32] DOI: 10.2307/1968202 · Zbl 0006.37101 · doi:10.2307/1968202
[33] DOI: 10.1090/S0002-9904-1941-07395-7 · Zbl 0025.23602 · doi:10.1090/S0002-9904-1941-07395-7
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