Chaos in hyperspace system

Let (X, d) be a compact metric space and f:X → X   be continuous. Let f¯ be the natural extension of f to the space of all non-empty compact subsets of X endowed with the Hausdorff metric induced by d. In this paper, some dynamical properties of f   and f¯ are considered. It is shown that positive topological entropy, Li–Yorke chaos and distributional chaos of f   imply those of f¯, respectively, but not conversely. The results give an answer to the question proposed by Román-Flores.