The mimimally displaced set of an irreducible automorphism is locally finite

We prove that the minimally displaced set of a relatively irreducible automorphism of a free splitting, situated in a deformationspace, is uniformly locally finite. The minimally displaced set coincideswith the train track points for an irreducible automorphism. We developthe theory in a general setting of deformation spaces of free products, having in mind the study of the action of reducible automorphisms of afree group on the simplicial bordification of Outer Space. For instance, a reducible automorphism will have invariant free factors, act on the corresponding stratum of the bordification, and in that deformation space it may be irreducible (sometimes this is referred as relative irreducibility).