Criteria for extension of commutativity to fractional iterates of holomorphic self-maps in the unit disc

Let $\phi$ be a univalent non-elliptic self-map of the unit disc $\mathbb{D}$ and let $({\psi }_t)$ be a continuous one-parameter semigroup of holomorphic functions in $\mathbb{D}$ such that ${\psi }_1\mathrm{\ne }{\mathrm{i}\mathrm{d}}_{\mathbb{D}}$ commutes with $\phi$. This assumption does not imply that all elements of the semigroup $({\psi }_t)$ commute with $\phi$. In this paper, we provide a number of sufficient conditions that guarantee that ${\psi }_t\circ \phi =\phi \circ {\psi }_t$ for all $t>0$: this holds, for example, if $\phi$ and ${\psi }_1$ have a common boundary (regular or irregular) fixed point different from their common Denjoy-Wolff point $\tau$, or when ${\psi }_1$ has a boundary regular fixed point $\sigma \mathrm{\ne }\tau$ at which $\phi$ is isogonal, or when $(\phi -{\mathrm{i}\mathrm{d}}_{\mathbb{D}})\mathrm{/}({\psi }_1-{\mathrm{i}\mathrm{d}}_{\mathbb{D}})$ has an unrestricted limit at $\tau$. In addition, we analyze how $\phi$ behaves in the petals of the semigroup $({\psi }_t)$.