A Conley-type decomposition of the strong chain recurrent set. (English) Zbl 1512.37014
Summary: For a continuous flow on a compact metric space, the aim of this paper is to prove a Conley-type decomposition of the strong chain recurrent set. We first discuss in detail the main properties of strong chain recurrent sets. We then introduce the notion of strongly stable set as a closed invariant set which is the intersection of the \(\omega\)-limits of a specific family of nested and definitively invariant neighborhoods of itself. This notion strengthens that of stable set; moreover, any attractor turns out to be strongly stable. We then show that strongly stable sets play the role of attractors in the decomposition of the strong chain recurrent set; indeed, we prove that the strong chain recurrent set coincides with the intersection of all strongly stable sets and their complementary.
MSC:
| 37B02 | Dynamics in general topological spaces |
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 37B20 | Notions of recurrence and recurrent behavior in topological dynamical systems |
| 37B30 | Index theory for dynamical systems, Morse-Conley indices |
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