Stability under powers of minset of hyperbolic irreducible automorphism

We show that in Outer Space, the minset of the displacement function of a hyperbolic irreducible automorphism eventually stabilizes under further powers if and only if no train track representative of the automorphism has a Pre-Nielsen Path. We then analyze what automorphisms of dierent ranks and indices have stable minsets, showing that almost every index of automorphism has examples with an eventually stable and never stable minset.