Topological entropy for induced hyperspace maps

Let (X, d) be a compact metric space and let f : X → X   be continuous. Let K(X)K(X) be the family of compact subsets of X   endowed with the Hausdorff metric and define the extension f¯:K(X)→K(X) by f¯(K)=f(K) for any K∈K(X)K∈K(X). We prove that the topological entropy of f¯ is greater or equal than the topological entropy of f, and this inequality can be strict. On the other hand, we prove that the topological entropy of f   is positive if and only if the topological entropy of f¯ is also positive.