Block-Coppels chaos in set-valued discrete systems

Let (X, d) be a compact metric space and f : X ? X be a continuous map. Consider the metric space (K(X),H) of all non empty compact subsets of X endowed with the Hausdorff metric induced by d. Let ¯ f : K(X) ? K(X) be defined by ¯ f(A) = {f(a) : a ? A} . We show that Block-Coppels chaos in f implies Block-Coppels chaos in ¯ f if f is a bijection.