Chaos on Hyperspace

In this paper, the chaotic behavior of a set-valued mapping F : X → 2 , where X is a compact space, is investigated. The existence of the generalized shadowing property in the hyperspace 2 is proved. Based on the generalized shadowing property of the set-valued mappings F and the assumption of the existence of an unstable chain recurrent point of the mapping F , it is shown that the Bernoulli system of bi-directional shifts is embedded in the sense of semiconjugacy into the image of mapping F , i.e. Smale’s chaos in the set-valued system F is thereby proved.