2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/163025We present a coarse convexity result for the dynamics of free group automorphisms: Given an automorphism $ϕ$ of a finitely generated free group $F$, we show that for all $x\in F$ and $0\leq i\leq N$, the length of $ϕ^i(x)$ is bounded above by a constant multiple of the sum of the lengths of $x$ and $ϕ^N(x)$, with the constant depending only on $ϕ$.35 pages, 6 figures; updated to reflect arXiv:0806.2889v1 in particularGroup TheoryGeometric Topology37E30text