The authors develop a notion of axis in the Culler–Vogtmann outer space \(\mathcal{X}_r\) of a finite rank free group \(F_r\), with respect to the action of a nongeometric, fully irreducible outer automorphism \(\phi\). Unlike the situation of a loxodromic isometry acting on hyperbolic space, or a pseudo-Anosov mapping class acting on Teichmüller space, \(\mathcal{X}_r\) has no natural metric, and \(\phi\) seems not to have a single natural axis. Instead these axes for \(\phi\), while not unique, fit into an “axis bundle” \(\mathcal{A}_\phi\) with nice topological properties: \(\mathcal{A}_\phi\) is a closed subset of \(\mathcal{X}_r\) proper homotopy equivalent to a line, it is invariant under \(\phi\), the two ends of \(\mathcal{A}_\phi\) limit on the repeller and attractor of the source–sink action of \(\phi\) on compactified outer space, and \(\mathcal{A}_\phi\) depends naturally on the repeller and attractor.

The authors propose various definitions for \(\mathcal{A}_\phi\), each motivated in different ways by train track theory or by properties of axes in Teichmüller space, and they prove their equivalence.