Topological entropy for induced hyperspace maps

Let (Xd) be a compact metric space and let f:X -> X be continuous. Let 1,(X) be the family of compact subsets of X endowed with the Hausdorff metric and define the extension (f) over bar: K(X) -> K(X) by (f) over bar (K) =f(K) for any K is an element of K(X). We prove that the topological entropy of I is greater or equal than the topological entropy off, and this inequality can be strict. On the other hand, we prove that the topological entropy off is positive if and only if the topological entropy of 7 is also positive. (c) 2005 Elsevier Ltd. All rights reserved.