  We develop a notion of axis in the Culler--Vogtmann outer space 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
Teichmuller space, X_r has no natural metric, and phi seems not to have a
single natural axis. Instead our axes for phi, while not unique, fit into an
``axis bundle'' A_phi with nice topological properties: A_phi is a closed
subset of X_r proper homotopy equivalent to a line, it is invariant under phi,
the two ends of A_phi limit on the repeller and attractor of the source--sink
action of phi on compactified outer space, and A_phi depends naturally on the
repeller and attractor. We propose various definitions for A_phi, each
motivated in different ways by train track theory or by properties of axes in
Teichmuller space, and we prove their equivalence.
