We show that every automorphism $\alpha$ of a free group $F_k$ of finite rank $k$ has {\it asymptotically periodic\/} dynamics on $F_k$ and its boundary $\partial F_k$: there exists a positive power $\alpha^q$ such that every element of the compactum $F_k \cup \partial F_k$ converges to a fixed point under iteration of $\alpha^q$.