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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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
- discrete dynamical system
- ω-limit set