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

The study of iterated functions is fundamental in complex dynamics. For a holomorphic self-map φ φ of the unit disk D D , the Denjoy–Wolff theorem (1926) establishes a key convergence property: if φ φ is not an elliptic automorphism, then its sequence of iterates, (φ ∘ n) ( φ ∘ n ) , converges uniformly on compact subsets to a point τ ∈ D̄ τ ∈ D ¯ , called the Denjoy–Wolff point of φ φ . A related question concerns the behavior of these iterates at the boundary. By Fatou’s theorem, all functions φ ∘ 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 (φ ∘ n)^* ( φ ∘ n ) ∗ . The behavior of the sequence ((φ ∘ n)^*) ( ( φ ∘ n ) ∗ ) depends significantly on whether φ φ is an inner function or not. When φ φ is an inner function, the behaviour of ((φ ∘ n)^*) ( ( φ ∘ n ) ∗ ) is well-established and can be found in texts by Aaronson or Doering and Mañé. However, when φ 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 φ