Invariant laminations for irreducible automorphisms of free groups

For every atoroidal iwip automorphism $\phi$ of $F_N$ (i.e. the analogue of a pseudo-Anosov mapping class) it is shown that the algebraic lamination dual to the forward limit tree $T_+(\phi)$ is obtained as "diagonal closure" of the support of the backward limit current $\mu_-(\phi)$. This diagonal closure is obtained through a finite procedure in analogy to adding diagonal leaves from the complementary components to the stable lamination of a pseudo-Anosov homeomorphism. We also give several new characterizations as well as a structure theorem for the dual lamination of $T_+(\phi)$, in terms of Bestvina-Feighn-Handel's "stable lamination" associated to $\phi$.