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On the structure of transitive ω-limit sets for continuous maps

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For continuous maps in compact metric spaces, the admissible topological structure of attractors of single trajectories is discussed. For transitive ω-limit sets, the admissible topological structure and the dynamics on their are components are described. Examples of sets with very simple structure, which fail to be ω-limit sets in ℝ2, are suggested. It is proved that for a complete characterization of ω-limit, sets in terms of are components, we should take into consideration not only the number of are components and their intersections but also the way in which convergence continua in the set are approximated.

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  1. 1.

    S. Agronsky, A. Bruckner, J. Ceder, T. Pearson,The structure of ω-limit sets for continuous functions. Real Analysis Exchange,15 (1989–90, 483–510.

  2. 2.

    S. Agronsky andJ. Ceder,What sets can be ω-limit sets in E n?. Real Analysis Exchange,17 (1991–92), 97–109.

  3. 3.

    S. Agronsky andJ. Ceder,Each Peano subspace of E k is an ω-limit set. Real Analysis Exchange,17 (1991–92), 371–378.

  4. 4.

    M. Babilonova,On a conjecture of Agronsky and Ceder concerning orbit-enclosing ω-limit sets. Real Anal. Exchange,23 (1997/98), no. 2, 773–777.

  5. 5.

    R. L. Devaney,Knaster-like continua and complex dynamics. Ergod. Th. & Dynam. Sys.,13 (1993), 627–634.

  6. 6.

    Y. N. Dowker andF. G. Friedlander,On limit sets in dynamical systems, Proc. London Math. Soc. (3)4 (1954), 168–176.

  7. 7.

    M. Handel,A pathological area preserving C diffeomorphism of the plane, Proc. A. M. S.248 (1982), 163–168.

  8. 8.

    V. Jiménez López andJ. Smítal,Two counterexamples to a conjecture by Agronsky and Ceder. Acta Math. Hungar.,88 (2000), no. 3, 193–204.

  9. 9.

    V. Jiménez López andJ. Smítal,ω-limit sets for triangular mappings. Fundamenta Mathematicae,167 (2001), 1–15.

  10. 10.

    J. Kennedy,Compactifying the space of homeomorphisms. Coll. Math.,56 (1988), 41–58.

  11. 11.

    S. B. Nadler, Jr.,Continuum Theory: an Introduction, Monographs and Textbooks in Pure and Applied Math.158, Marcel Dekker, Inc., New York, Basel, Hong Kong (1992).

  12. 12.

    V. V. Nemytskii andV. V. Stepanov,Qualitative Theory of Differential Equations. Princeton Univ. Press: Princeton, N.J., 1960.

  13. 13.

    A. N. Šarkovskii,Attracting and attracted sets. Dokl. Akad. Nauk SSSR160 (1965), 1036–1038.English translation in Sov. Math. Dokl.6 (1965), 268–270.

  14. 14.

    A. G. Sivak,Each nowhere dense nonvoid closed set inn is a δ-limit set. Fundamenta Mathematicae149 (1996), 183–190.

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Author information

Correspondence to Andrei G. Sivak.

Additional information

Supported in part by State Foundation for Fundamental Research of the Ministry of Education and Science of Ukraine, project No. 01.07/00081.

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Sivak, A.G. On the structure of transitive ω-limit sets for continuous maps. Qual. Th. Dyn. Syst. 4, 109 (2003). https://doi.org/10.1007/BF02972825

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Key Words

  • discrete dynamical system
  • ω-limit set