Topological entropy for induced hyperspace maps

Let (X, d) be a compact metric space and let f : X → X be continuous. Let be the family of compact subsets of X endowed with the Hausdorff metric and define the extension by for any . We prove that the topological entropy of 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 is also positive.