The study of iterated functions is fundamental in complex dynamics. For a holomorphic self-map \(\varphi \) of the unit disk \(\mathbb {D}\) , the Denjoy–Wolff theorem (1926) establishes a key convergence property: if \(\varphi \) is not an elliptic automorphism, then its sequence of iterates, \((\varphi ^{\circ n})\) , converges uniformly on compact subsets to a point \(\tau \in \overline{\mathbb {D}}\) , called the Denjoy–Wolff point of \(\varphi \) . A related question concerns the behavior of these iterates at the boundary. By Fatou’s theorem, all functions \(\varphi ^{\circ n}\) have non-tangential limits at almost every point on the boundary of the unit disk. We denote these non-tangential limits as \((\varphi ^{\circ n})^*\) . The behavior of the sequence \(((\varphi ^{\circ n})^*)\) depends significantly on whether \(\varphi \) is an inner function or not. When \(\varphi \) is an inner function, the behaviour of \(((\varphi ^{\circ n})^*)\) is well-established and can be found in texts by Aaronson or Doering and Mañé. However, when \(\varphi \) 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 \(\varphi \) is not an inner function, then the sequence \(((\varphi ^{\circ n})^*)\) converges to \(\tau \) for almost every point on \(\partial \mathbb {D}\) . Our technique also provides general results on non-autonomous iteration of holomorphic self-maps of the unit disk.
