Equivalent conditions of Devaney chaos on the hyperspace

Let T be a continuous self-map of a compact metric space X. The transformation T induces naturally a continuous self-map TK on the hyperspace K(X) of all non-empty closed subsets of X. It is shown that the system (K(X),TK) is Devaney chaotic if and only if (K(X),TK) is an HY-system if and only if (X,T) is an HY-system, where a system is called an HY-system if it is totally transitive and has dense small periodic sets.