The free shift $\alpha$ of the reduced free product $C^*$-algebra $A$ is studied from both the analytic and non-commutative ergodic theoretic viewpoints. For an automorphism $\beta$ of $B$, we show that the entropy of $\mathop{\rm Ad}\nolimits u(\alpha \otimes \beta)$ is equal to the entropy of $\mathop{\rm Ad}\nolimits u(\beta)$. We also show that if $B$ is unital, nuclear, and simple, and if the crossed product $B \rtimes_\beta {\Bbb Z}$ is simple and purely infinite, then $(O_\infty \otimes B)\rtimes_{\alpha \otimes \beta} {\Bbb Z}$ is isomorphic to $B \rtimes_\beta {\Bbb Z}$.