Kato's chaos in set-valued discrete systems

In this paper, 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: X → X 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 we-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.