On indecomposable trees in the boundary of Outer space

Let $T$ be an $\mathbb{R}$-tree, equipped with a very small action of the rank $n$ free group $F_n$, and let $H \leq F_n$ be finitely generated. We consider the case where the action $F_n \curvearrowright T$ is indecomposable--this is a strong mixing property introduced by Guirardel. In this case, we show that the action of $H$ on its minimal invarinat subtree $T_H$ has dense orbits if and only if $H$ is finite index in $F_n$. There is an interesting application to dual algebraic laminations; we show that for $T$ free and indecomposable and for $H \leq F_n$ finitely generated, $H$ carries a leaf of the dual lamination of $T$ if and only if $H$ is finite index in $F_n$. This generalizes a result of Bestvina-Feighn-Handel regarding stable trees of fully irreducible automorphisms.