Topological entropy for set valued maps

Any continuous map T on a compact metric space X induces in a natural way a continuous map (T) over bar on the space K(X) of all non-empty compact subsets of X. Let T be a homeomorphism on the interval or on the circle. It is proved that the topological entropy of the induced set valued map (T) over bar is zero or infinity. Moreover, the topological entropy of (T) over bar vertical bar(C(X)) is zero, where C(X) denotes the space of all non-empty compact and connected subsets of X. For general continuous maps on compact metric spaces these results are not valid. Crown Copyright (C) 2010 Published by Elsevier Ltd. All rights reserved.