Topological Entropy of Free Product Automorphisms
Nathanial P. Brown, Kenneth Dykema, and Dimitri Shlyakhtenko
Using free probability constructions involving the Cuntz–Pimsner $C^{*}$–algebra we show that the topological entropy ${\rm ht} (\alpha *\beta)$ of the free product of two automorphisms is given by the maximum $\max ({\rm ht} (\alpha ),{\rm ht} (\beta ))$. As applications, we show in full generality that free shifts have topological entropy zero. We show that any separable nuclear $C^{*}$–dynamical system can be covariantly embedded into $\mathcal{O}_{2}$ and $\mathcal{O}_{\infty}$ in an entropy–preserving way. It follows that any nuclear simple purely infinite $C^{*}$–algebra admits an automorphism with any given value of entropy. We also show that the free product of two automorphisms satisfies the Connes–Narnhofer–Thirring variational principle, if the two automorphisms do.