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 natural a continuous self-map $T_K$ on the hyperspace $K(X)$ of all non-empty closed subsets of $X$. In this paper, we show that the system $(K(X),T_K)$ on the hyperspace is Devaney chaotic if and only if $(K(X),T_K)$ is an HY-system if and only if $(X,T)$ is an HY-system, where a system $(Y,S)$ is called an HY-system if it is totally transitive and has dense small periodic sets.