Approximation entropies in crossed products with an application to free shifts

It is shown that for two dynamical approximation entropies (one C* and one W*) the implementing inner automorphism in a crossed product A ⋊_α ℤ has the same entropy value as the automorphism α. Using the techniques in the proof, an example of a highly ergodic non-asymptotically abelian automorphism with topological entropy zero is also given. More specifically, it is shown that the free shifts on the Cuntz algebra script O sign_∞ and the reduced free group C*-algebra C* (double-struck F sign_∞) have topological entropy zero.