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 Teichmueller 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 Teichmueller space, and they prove their equivalence.
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