Chaos for induced hyperspace maps

For (X, d) be a metric space, f : X → X a continuous map and (K(X),H) the space of non-empty compact subsets of X with the Hausdorff metric, one may study the dynamical properties of the induced map (∗)f¯:K(X)→K(X):A↦f(A).H. Román-Flores [A note on in set-valued discrete systems. Chaos, Solitons & Fractals 2003;17:99-104] has shown that if f¯ is topologically transitive then so is f, but that the reverse implication does not hold. This paper shows that the topological transitivity of f¯ is in fact equivalent to weak topological mixing on the part of f. This is proved in the more general context of an induced map on some suitable hyperspace H of X with the Vietoris topology (which agrees with the topology of the Hausdorff metric in the case discussed by Román-Flores.