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Kato’s chaos in set-valued discrete systems. (English) Zbl 1140.37305

Summary: We investigate the relationships between Kato’s chaoticity of a dynamical system \((X, f)\) and Kato’s chaoticity of the set-valued discrete system \((\mathcal K(X),\bar f)\) associated to \((X, f)\), where \(X\) is a compact metric space and \(f: X \rightarrow X\) is a continuous map. We show that Kato’s chaoticity of \((\mathcal K(X),\bar f)\) implies the Kato’s chaoticity of \((X, f)\) in general and \((X, f)\) is chaotic in the sense of Kato if and only if \((\mathcal K(X),\bar f)\) is Kato chaotic in \(w^e\)-topology. We also show that Ruelle-Takens’ chaoticity implies Kato’s chaoticity for a continuous map with a fixed point from a complete metric space without isolated point into itself.

MSC:

37B99 Topological dynamics
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