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

Abstract

Let $φ$ be a univalent non-elliptic self-map of the unit disc $\mathbb D$ and let $(ψ_{t})$ be a continuous one-parameter semigroup of holomorphic functions in $\mathbb D$ such that $ψ_{1}\neq\mathrm{id}_{\mathbb D}$ commutes with $φ$. This assumption does not imply that all elements of the semigroup $(ψ_t)$ commute with $φ$. In this paper, we provide a number of sufficient conditions that guarantee that ${ψ_t\circφ=φ\circψ_t}$ for all ${t>0}$: this holds, for example, if $φ$ and $ψ_1$ have a common boundary (regular or irregular) fixed point different from their common Denjoy-Wolff point $τ$, or when $ψ_1$ has a boundary regular fixed point ${σ\neqτ}$ at which $φ$ is isogonal, or when $(φ-\mathrm{id}_{\mathbb D})/(ψ_1-\mathrm{id}_{\mathbb D})$ has an unrestricted limit at $τ$. In addition, we analyze how $φ$ behaves in the petals of the semigroup $(ψ_t)$.