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 D and let (psi(t)) be a continuous one-parameter semigroup of holomorphic functions in D such that psi(1)not equal id(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)degrees phi=phi degrees 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 not equal tau at which phi is isogonal, or when (phi-idD)/(psi(1)-id(D)) has an unrestricted limit at tau. In addition, we analyze how phi behaves in the petals of the semigroup (psi(t)).