Paper detail

Dynamics of Irreducible Endomorphisms of $F_n$

We consider the class non-surjective irreducible endomorphisms of the free group $F_n$. We show that such an endomorphism $ϕ$ is topologically represented by a simplicial immersion $f:G \rightarrow G$ of a marked graph $G$; along the way we classify the dynamics of $\partial ϕ$ acting on $\partial F_n$: there are at most $2n$ fixed points, all of which are attracting. After imposing a necessary additional hypothesis on $ϕ$, we consider the action of $ϕ$ on the closure $\bar{CV}_n$ of the Culler-Vogtmann Outer space. We show that $ϕ$ acts on $\bar{CV}_n$ with "sink" dynamics: there is a unique fixed point $[T_ϕ]$, which is attracting; for any compact neighborhood $N$ of $[T_ϕ]$, there is $K=K(N)$, such that $\bar{CV}_nϕ^{K(N)} \subseteq N$. The proof uses certian projections of trees coming from invariant length measures. These ideas are extended to show how to decompose a tree $T$ in the boundary of Outer space by considering the space of invariant length measures on $T$; this gives a decomposition that generalizes the decomposition of geometric trees coming from Imanishi's theorem.