Dynamics of free group automorphisms

We present a coarse convexity result for the dynamics of free group automorphisms: Given an automorphism $\varphi$ of a finitely generated free group $F$, we show that for all $x\in F$ and $0\le i\le N$, the length of ${\varphi }^i(x)$ is bounded above by a constant multiple of the sum of the lengths of $x$ and ${\varphi }^N(x)$, with the constant depending only on $\varphi$.