Handel and Mosher define the axis bundle for a fully irreducible outer automorphism in [Mem. Amer. Math. Soc. 1004 (2011)]. We give a necessary and sufficient condition for the axis bundle to consist of a unique periodic fold line. As a consequence, we give a setting, and means for identifying in this setting, when two elements of an outer automorphism group have conjugate powers.
"Lone axes in outer space." Algebr. Geom. Topol. 16 (6) 3385 - 3418, 2016. https://doi.org/10.2140/agt.2016.16.3385