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

Abstract Let be a univalent non‐elliptic self‐map of the unit disc and let be a continuous one‐parameter semigroup of holomorphic functions in such that commutes with . This assumption does not imply that all elements of the semigroup commute with . In this paper, we provide a number of sufficient conditions that guarantee that for all : This holds, for example, if and have a common boundary (regular or irregular) fixed point different from their common Denjoy–Wolff point , or when has a boundary regular fixed point at which is isogonal, or when has an unrestricted limit at . In addition, we analyze how behaves in the petals of the semigroup .