2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/219679Our goal is to find dynamic invariants that completely determine elements of the outer automorphism group $\Out(F_n)$ of the free group $F_n$ of rank $n$. To avoid finite order phenomena, we do this for {\it forward rotationless} elements. This is not a serious restriction. For example, there is $K_n>0$ depending only on $n$ such that, for all $ϕ\in\Out(F_n)$, $ϕ^{K_n}$ is forward rotationless. An important part of our analysis is to show that rotationless elements are represented by particularly nice relative train track maps.68 pagesGroup Theory20F65; 20F28text