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This is the third in a series of four papers (with research announcementposted on this arXiv) that develop a decomposition theory for subgroups of$\text{Out}(F_n)$.  In this paper, given an outer automorphism of $F_n$ and anattracting-repelling lamination pair, we study which lines and conjugacyclasses in $F_n$ are weakly attracted to that lamination pair under forward andbackward iteration respectively. For conjugacy classes, we prove Theorem F fromthe research annoucement, which exhibits a unique vertex group system calledthe "nonattracting subgroup system" having the property that the conjugacyclasses it carries are characterized as those which are not weakly attracted tothe attracting lamination under forward iteration, and also as those which arenot weakly attracted to the repelling lamination under backward iteration. Forlines in general, we prove Theorem G that characterizes exactly which lines areweakly attracted to the attracting lamination under forward iteration and whichto the repelling lamination under backward iteration. We also prove Theorem Hwhich gives a uniform version of weak attraction of lines.  v3: Contains a stronger proof of Lemma 2.19 (part of the proof of Theorem G)for purposes of further applications.

Subgroup decomposition in $\text{Out}(F_n)$, Part III: Weak attraction theory