Entropy for automorphisms of free groups

Let $\sigma$ be the automorphism of the free group $F_\infty$ which is arising from a permutation of the free generators of $F_\infty.$ The $\sigma$ naturally induces the automorphism $\hat \sigma$ of the reduced $C^*$-algebra $C^*_r(F_\infty),$ and also the automorphism $\bar{\hat \sigma}$ of the group factor $L(F_\infty).$ We show that the Brown-Germain entropy $ha(\sigma )$ is zero. This implies that the Brown-Voiculescu topological entropy $ht(\hat \sigma ),$ the Connes-Narnhofer-Thirring dynamical entropy $h_\phi (\hat \sigma )$ and the Connes-Størmer entropy $H(\bar{\hat \sigma} )$ are all zero.