TOPOLOGICAL ENTROPY OF FREE PRODUCT AUTOMORPHISMS

NATHANIAL P. BROWN KENNETH DYKEMA DIMITRI SHLYAKHTENKO

Abstract. Using free probability constructions involving the Cuntz–Pimsner C^{∗}– algebra we show that the topological entropy ht(α ∗ β) of the free product of two automorphisms is given by the maximum max(ht(α), ht(β)). 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 O_{2} and O_{∞} 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.