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

Let φ be a univalent non-elliptic self-map of the unit disc D and let (ψₜ) be a continuous one-parameter semigroup of holomorphic functions in D such that ψ₁D commutes with φ. This assumption does not imply that all elements of the semigroup (ψₜ) commute with φ. In this paper, we provide a number of sufficient conditions that guarantee that ψₜ∘φ=φ∘ψₜ for all t>0: this holds, for example, if φ and ψ₁ have a common boundary (regular or irregular) fixed point different from their common Denjoy-Wolff point τ, or when ψ₁ has a boundary regular fixed point σ≠τ at which φ is isogonal, or when (φ-idD)/(ψ₁-idD) has an unrestricted limit at τ. In addition, we analyze how φ behaves in the petals of the semigroup (ψₜ).