Approximation entropies in crossed products with an application to free shifts

Nathanial P. Brown and Marie Choda

Vol. 198 (2001), No. 2, 331–346

DOI: 10.2140/pjm.2001.198.331

Abstract

It is shown that for two dynamical approximation entropies (one C^∗ and one W^∗) the implementing inner automorphism in a crossed product A ⋊ _αℤ 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 𝒪_∞ and the reduced free group C^∗-algebra C_r^∗(𝔽_∞) have topological entropy zero.

Milestones

Received: 15 July 1999

Revised: 6 April 2000

Published: 1 April 2001