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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 (𝒦(X),f ¯) associated to (X,f), where X is a compact metric space and f:XX is a continuous map. We show that Kato’s chaoticity of (𝒦(X),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 (𝒦(X),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:
37B99Topological dynamics