Topological Entropy of Free Product Automorphisms

TL;DR

This paper demonstrates that the topological entropy of the free product of two automorphisms equals the maximum of their individual entropies, with applications to free shifts, nuclear C*-dynamical systems, and automorphism entropy values.

Contribution

It introduces a novel approach using free probability and Cuntz-Pimsner C*-algebras to analyze entropy in free product automorphisms, establishing new entropy relations and embeddings.

Findings

Free product automorphisms have entropy equal to the maximum of individual entropies.

General free shifts have zero entropy.

Any nuclear C*-dynamical system can embed into the Cuntz algebra preserving entropy.

Abstract

Using free probability constructions involving Cuntz-Pimsner C*-algebras we show that the topological entropy of the free product of two automorphisms is equal to the maximum of the individual entropies. As applications we show that general free shifts have entropy zero. We show that any nuclear C*-dynamical system admits an entropy preserving covariant embedding into the Cuntz algebra on infinitely many generators. It follows that any simple nuclear purely infinite C*-algebra admits an automorphism with any given value of entropy. As a final application we show that if two automorphisms satisfy a CNT-variational principle then so does their free product.