### Free-group automorphisms, train tracks, and the Beaded Decomposition

#### Submitted

We study the automorphisms $\phi$ of a finitely generated free group $F$. Building on the train-track technology of Bestvina, Feighn and Handel, we provide a topological representative $f:G\to G$ of a power of $\phi$ that behaves very much like the realization on the rose of a positive automorphism. This resemblance is encapsulated in the {\em Beaded Decomposition Theorem} which describes the structure of paths in $G$ obtained by repeatedly passing to $f$-images of an edge and taking subpaths. This decomposition is the key to adapting our proof of the quadratic isoperimetric inequality for $F\rtimes_\phi\mathbb Z$, with $\phi$ positive, to the general case.