Topological entropy for set valued maps

Any continuous map TT on a compact metric space XX induces in a natural way a continuous map T¯ on the space K(X)K(X) of all non-empty compact subsets of XX. Let TT be a homeomorphism on the interval or on the circle. It is proved that the topological entropy of the induced set valued map T¯ is zero or infinity. Moreover, the topological entropy of T¯|C(X) is zero, where C(X)C(X) denotes the space of all non-empty compact and connected subsets of XX. For general continuous maps on compact metric spaces these results are not valid.