  Let $G$ be a group and let ${\mathcal G}$ be a free factor system of $G$,
namely a free splitting of $G$ as $G=G_1*\dots*G_k*F_r$. In this paper, we
study the set of train track points for ${\mathcal G}$-irreducible
automorphisms $\phi$ with exponential growth (relatively to ${\mathcal G}$).
Such set is known to coincide with the minimally displaced set
$\operatorname{Min}(\phi)$ of $\phi$.
  Our main result is that $\operatorname{Min}(\phi)$ is co-compact, under the
action of the cyclic subgroup generated by $\phi$.
  Along the way we obtain other results that could be of independent interest.
For instance, we prove that any point of $\operatorname{Min}(\phi)$ is in
uniform distance from $\operatorname{Min}(\phi^{-1})$. We also prove that the
action of $G$ on the product of the attracting and the repelling trees for
$\phi$, is discrete. Finally, we get some fine insight about the local topology
of relative outer space.
  As an application, we generalise a classical result of Bestvina, Feighn and
Handel for the centralisers of irreducible automorphisms of free groups, in the
more general context of relatively irreducible automorphisms of a free product.
We also deduce that centralisers of elements of $\operatorname{Out}(F_3)$ are
finitely generated, which was previously unknown. Finally, we mention that an
immediate corollary of co-compactness is that $\operatorname{Min}(\phi)$ is
quasi-isometric to a line.
