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

Abstract

Let $\phi$ be a univalent non-elliptic self-map of the unit disc $D$ and let $\left({\psi }_t\right)$ be a continuous one-parameter semigroup of holomorphic functions in $D$ such that ${\psi }_1\ne {\mathrm{id}}_D$ commutes with $\phi$. This assumption does not imply that all elements of the semigroup $\left({\psi }_t\right)$ 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 $\sigma \ne \tau$ at which $\phi$ is isogonal, or when $(\phi -{\mathrm{id}}_D)/({\psi }_1-{\mathrm{id}}_D)$ has an unrestricted limit at $\tau$. In addition, we analyze how $\phi$ behaves in the petals of the semigroup $\left({\psi }_t\right)$.