Normalizers and centralizers of cyclic subgroups generated by lone axis fully irreducible outer automorphisms

We let $\varphi$ be an ageometric fully irreducible outer automorphism so that its Handel-Mosher axis bundle consists of a single unique axis. We show that the centralizer $Cen(\langle\varphi\rangle)$ of the cyclic subgroup generated by $\varphi$ equals the stabilizer $\text{Stab}(\Lambda^+_\varphi)$ of the attracting lamination $\Lambda^+_{\varphi}$ and is isomorphic to $\mathbb Z$. We further show, via an analogous result about the commensurator, that the normalizer $N(\langle\varphi\rangle)$ of $\langle \varphi \rangle$ is isomorphic to either $\mathbb Z$ or $\mathbb Z_2 * \mathbb Z_2$.