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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, westudy the set of train track points for ${\mathcal G}$-irreducibleautomorphisms $\phi$ 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 theaction 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 inuniform distance from $\operatorname{Min}(\phi^{-1})$. We also prove that theaction 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 topologyof relative outer space.  As an application, we generalise a classical result of Bestvina, Feighn andHandel for the centralisers of irreducible automorphisms of free groups, in themore general context of relatively irreducible automorphisms of a free product.We also deduce that centralisers of elements of $\operatorname{Out}(F_3)$ arefinitely generated, which was previously unknown. Finally, we mention that animmediate corollary of co-compactness is that $\operatorname{Min}(\phi)$ isquasi-isometric to a line.

On the action of relatively irreducible automorphisms on their train tracks