Robinson's chaos in set-valued discrete systems

Let (X, d) be a compact metric space and f : X → X a continuous function. If we consider the space (K(X),H) of all non-empty compact subsets of X endowed with the Hausdorff metric induced by d and f¯:K(X)→K(X), f¯(A)={f(a)/a∈A}, then the aim of this work is to show that Robinson's chaos in f¯ implies Robinson's chaos in f. Also, we give an example showing that R-chaos in f does not implies R-chaos in f¯.