## 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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### References:

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