Dynamics of Irreducible Endomorphisms of $F_n$

Abstract

We consider the class non-surjective irreducible endomorphisms of the free group $F_n$. We show that such an endomorphism $\varphi$ is topologically represented by a simplicial immersion $f:G\to G$ of a marked graph $G$; along the way we classify the dynamics of $\partial \varphi$ acting on $\partial F_n$: there are at most $2n$ fixed points, all of which are attracting. After imposing a necessary additional hypothesis on $\varphi$, we consider the action of $\varphi$ on the closure ${\overline{CV}}_n$ of the Culler-Vogtmann Outer space. We show that $\varphi$ acts on ${\overline{CV}}_n$ with "sink" dynamics: there is a unique fixed point $[T_{\varphi }]$, which is attracting; for any compact neighborhood $N$ of $[T_{\varphi }]$, there is $K=K(N)$, such that ${\overline{CV}}_n{\varphi }^{K(N)}\subseteq N$. 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.

arXiv commentv3, 46 pages, corrected gap in decomposition result

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• Cubulating hyperbolic free-by-cyclic groups: the irreducible case
• The boundary of the free splitting graph and the free factor graph
• The dynamics and geometry of free group endomorphisms
• Surface subgroups from linear programming
• Cannon-Thurston fibers for iwip automorphisms of $F_N$
• Constructing non-uniquely ergodic arational trees
• Cubulating mapping tori of some polynomial growth free group automorphisms
• Detecting fully irreducible automorphisms: a polynomial time algorithm. With an appendix by Mark C. Bell