Abstract
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 dual to the forward limit tree $T_+(\phi)$ is obtained as 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 of a pseudo-Anosov homeomorphism. We also give several new characterizations as well as a structure theorem for the dual of $T_+(\phi)$, in terms of Bestvina-Feighn-Handel's stable lamination associated to $\phi$.
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