Localization of the chain recurrent set using shape theory and symbolical dynamics. (English) Zbl 1483.37026
Summary: The main aim of this paper is localization of the chain recurrent set in shape theoretical framework. Namely, using the intrinsic approach to shape from [N. Shekutkovski, Topol. Proc. 39, 27–39 (2012; Zbl 1215.54008)] we present a result which claims that under certain conditions the chain recurrent set preserves local shape properties. We proved this result in [N. Shekutkovski and M. Shoptrajanov, Topology Appl. 202, 117–126 (2016; Zbl 1341.54021)] using the notion of a proper covering. Here we give a new proof using the Lebesque number for a covering and verify this result by investigating the symbolical image of a couple of systems of differential equations following [G. Osipenko, Differ. Uravn. Protsessy Upr. 1998, No. 4, 59–74 (1998; Zbl 1488.37005)].
MSC:
| 37B20 | Notions of recurrence and recurrent behavior in topological dynamical systems |
| 37B35 | Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems |
| 37B25 | Stability of topological dynamical systems |
| 54C56 | Shape theory in general topology |
Keywords:
chain recurrent set; shape; intrinsic shape; symbolical image; attractor; Morse decomposition; Lyapunov functionReferences:
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