Low complexity among principal fully irreducible elements of Out($F_3$)

We find the shortest realized stretch factor for a fully irreducible $\phi \in \mathrm{O}\mathrm{u}\mathrm{t}(F_3)$ and show that it is realized by a “principal” fully irreducible element. We also show that it is the only principal fully irreducible produced by a single fold in any rank.