The Denjoy-Wolff theorem in simply connected domains

We characterize the simply connected domains S2 C C that exhibit the Denjoy-Wolff Property, meaning that every holomorphic self-map of S2 without fixed points has a Denjoy-Wolff point. We demonstrate that this property holds if and only if every automorphism of S2 without fixed points in S2 has a Denjoy-Wolff point. Furthermore, we establish that the Denjoy-Wolff Property is equivalent to the existence of what we term an "H-limit" at each boundary point for a Riemann map associated with the domain. The H-limit condition is stronger than the existence of non-tangential limits but weaker than unrestricted limits. As an additional result of our work, we prove that there exists bounded simply connected domains where the Denjoy-Wolff Property holds but which are not visible in the sense of Bharali and Zimmer. Since visibility is a sufficient condition for the Denjoy-Wolff Property, this proves that in general it is not necessary.