# The minimally displaced set of an irreducible automorphism is locally finite

## TL;DR

This paper proves that the set of points minimally displaced by a relatively irreducible automorphism in a deformation space of free splittings is uniformly locally finite, extending the understanding of automorphism actions on Outer Space.

## Contribution

It establishes the local finiteness of the minimally displaced set for relatively irreducible automorphisms within deformation spaces, generalizing train track theory.

## Key findings

- Minimally displaced set is uniformly locally finite.
- Minimally displaced set coincides with train track points.
- Applicable to deformation spaces of free products.

## Abstract

We prove that the minimally displaced set of a relatively irreducible automorphism of a free splitting, situated in a deformation space, is uniformly locally finite. The minimally displaced set coincides with the train track points for an irreducible automorphism. We develop the theory in a general setting of deformation spaces of free products, having in mind the study of the action of reducible automorphisms of a free 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).