Parageometric outer automorphisms of free groups

We study those fully irreducible outer automorphisms $\phi$ of a finite rank free group $F_r$ which are parageometric, meaning that the attracting fixed point of $\phi$ in the boundary of outer space is a geometric $\mathbf{R}$-tree with respect to the action of $F_r$, but $\phi$ itself is not a geometric outer automorphism in that it is not represented by a homeomorphism of a surface. Our main result shows that the expansion factor of $\phi$ is strictly larger than the expansion factor of $\phi^{-1}$. As corollaries (proved independently by Guirardel), the inverse of a parageometric outer automorphism is neither geometric nor parageometric, and a fully irreducible outer automorphism $\phi$ is geometric if and only if its attracting and repelling fixed points in the boundary of outer space are geometric $\mathbf{R}$-trees.