  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.
