Nathanial P. Brown, Kenneth Dykema, and Dimitri Shlyakhtenko
\newcommand{\HT}{\operatorname{ht}}
Using free probability constructions involving the Cuntz--Pimsner \(C^{*} \)--algebra we show that the topological entropy \( \HT (\alpha *\beta ) \) of the free product of two automorphisms is given by the maximum \( \max (\HT (\alpha ),\HT (\beta )) \). 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.