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Chaos in hyperspace system. (English) Zbl 1197.37016

Summary: Let \((X, d)\) be a compact metric space and \(f:X \rightarrow X\) be continuous. Let \(\overline{f}\) be the natural extension of \(f\) to the space of all non-empty compact subsets of \(X\) endowed with the Hausdorff metric induced by \(d\). In this paper, some dynamical properties of \(f\) and \(\overline{f}\) are considered. It is shown that positive topological entropy, Li-Yorke chaos and distributional chaos of \(f\) imply those of \(\overline{f}\), respectively, but not conversely. The results give an answer to the question proposed by Román-Flores.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:

37B40 Topological entropy
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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References:

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