Strongly contracting geodesics in Outer Space

Abstract

We study the Lipschitz metric on Outer Space and prove that fully irreducible elements of $Out\left(F_n\right)$ act by hyperbolic isometries with axes which are strongly contracting. As a corollary, we prove that the axes of fully irreducible automorphisms in the Cayley graph of $Out\left(F_n\right)$ are Morse, meaning that a quasi-geodesic with endpoints on the axis stays within a bounded distance from the axis.