Displacements of automorphisms of free groups I: Displacement functions, minpoints and train tracks

This is the first of two papers in which we investigate properties displacement functions automorphisms free groups (more generally, products) on Culler-Vogtmann Outer space and its simplicial bordification – splitting complex with respect to Lipschitz metric. The theory for irreducible being well-developed, concentrate reducible case. Since deal bordification, develop all needed tools more general setting deformation spaces, their associated complexes. In present paper study local function. particular, convexity behaviour at points, by geometrically characterising continuity-points. We prove that global-simplex-displacement spectrum A u t ( F n stretchy="false">) Aut(F_n) a well-ordered subset alttext="double-struck R"> R encoding="application/x-tex">\mathbb {R} , this helpful algorithmic purposes. introduce weaker notion train tracks, call partial tracks (which coincides usual one automorphisms) that, any automorphism, points minimal minpoints coincide marked metric graphs support partial tracks. show or not, has track (hence minpoint) either outer bordification. given an invariant factors seen map. subsequent will level sets are connected, apply result solve certain decision problems.