Growth and displacement of free product automorphisms

It is well known for an irreducible free group automorphism that its growth rate is equal to the minimal Lipschitz displacement of its action on Culler-Vogtmann space. This follows as a consequence of the existence of train track representatives for the automorphism. We extend this result to the general - possibly reducible - case as well as to the free product situation where growth is replaced by `relative growth’.