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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 $(\cal K(X),\bar f)$ 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 $(\cal K(X),\bar f)$ implies the Kato’s chaoticity of $(X, f)$ in general and $(X, f)$ is chaotic in the sense of Kato if and only if $(\cal K(X),\bar f)$ 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.

37B99Topological dynamics
Full Text: DOI
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