2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/170675We show that every automorphism $α$ 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 $α^q$ such that every element of the compactum $F_k \cup \partial F_k$ converges to a fixed point under iteration of $α^q$.Final version. To appear in Crelle's journalGroup TheoryGeometric Topologytext