Convergence on the boundary for iterates of holomorphic self-maps of the unit disk

Abstract The study of iterated functions is fundamental in complex dynamics. For a holomorphic self-map φ of the unit disk D D, the Denjoy–Wolff theorem (1926) establishes a key convergence property: if φ is not an elliptic automorphism, then its sequence of iterates, (^ n) (φ ∘ n), converges uniformly on compact subsets to a point D τ ∈ D ¯, called the Denjoy–Wolff point of φ. A related question concerns the behavior of these iterates at the boundary. By Fatou’s theorem, all functions ^ n φ ∘ n have non-tangential limits at almost every point on the boundary of the unit disk. We denote these non-tangential limits as (^ n) ^* (φ ∘ n) ∗. The behavior of the sequence ( (^ n) ^*) ( (φ ∘ n) ∗) depends significantly on whether φ is an inner function or not. When φ is an inner function, the behaviour of ( (^ n) ^*) ( (φ ∘ n) ∗) is well-established and can be found in texts by Aaronson or Doering and Mañé. However, when φ is not an inner function, the problem was not solved. Previous partial results have been contributed by Bourdon, Matache, and Shapiro; by Poggi-Corradini; and by Contreras, Díaz-Madrigal, and Pommerenke. In this paper, we achieved the final solution: if