Dynamics of irreducible endomorphisms of F(N)

We consider the class non-surjective irreducible endomorphisms of the free group Fn. We show that such an endomorphism &phis; is topologically represented by a simplicial immersion f : G → G of a marked graph G; along the way we classify the dynamics of ∂&phis; acting on ∂Fn: there are at most 2n fixed points, all of which are attracting. After imposing a necessary additional hypothesis on &phis;, we consider the action of &phis; on the closure CVn of the Culler-Vogtmann Outer space. We show that &phis; acts on CVn with "sink" dynamics: there is a unique fixed point [ T&phis;], which is attracting; for any compact neighborhood N of [T&phis;], there is K = K(N), such that CV n&phis;K(N ) ⊆ 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.;The proof of our main dynamics result uses a result of independent interest regarding certain actions in the boundary of Outer space. Let T be an R -tree, equipped with a very small action of the rank n free group Fn, and let H ≤ Fn be finitely generated. We consider the case where the action Fn ↷ T is indecomposable---this is a strong mixing property introduced by Guirardel. In this case, we show that the action of H on its minimal invariant subtree TH has dense orbits if and only if H is finite index in Fn. There is an interesting application to dual algebraic laminations; we show that for T free and indecomposable and for H ≤ Fn finitely generated, H carries a leaf of the dual lamination of T if and only if H is finite index in Fn. This generalizes a result of Bestvina-Feighn-Handel regarding stable trees of fully irreducible automorphisms.