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

We let $\phi$ 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 $Cen(⟨\phi ⟩)$ of the cyclic subgroup generated by $\phi$ equals the stabilizer $\text{Stab}({\mathrm{\Lambda }}_{\phi }^{+})$ of the attracting lamination ${\mathrm{\Lambda }}_{\phi }^{+}$ and is isomorphic to $\mathbb{Z}$. We further show, via an analogous result about the commensurator, that the normalizer $N(⟨\phi ⟩)$ of $⟨\phi ⟩$ is isomorphic to either $\mathbb{Z}$ or ${\mathbb{Z}}_2\ast {\mathbb{Z}}_2$.