This is the third in a series of four papers (with research announcement posted 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 an attracting-repelling lamination pair, we study which lines and conjugacy classes in $F_n$ are weakly attracted to that lamination pair under forward and backward iteration respectively. For conjugacy classes, we prove Theorem F from the research annoucement, which exhibits a unique vertex group system called the nonattracting subgroup system having the property that the conjugacy classes it carries are characterized as those which are not weakly attracted to the attracting lamination under forward iteration, and also as those which are not weakly attracted to the repelling lamination under backward iteration. For lines in general, we prove Theorem G that characterizes exactly which lines are weakly attracted to the attracting lamination under forward iteration and which to the repelling lamination under backward iteration. We also prove Theorem H which 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.
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