The Denjoy-Wolff Theorem in simply connected domains

TL;DR

This paper characterizes simply connected domains with the Denjoy-Wolff Property, linking it to automorphisms and boundary limits, and shows that visibility is not necessary for this property.

Contribution

It provides a new characterization of the Denjoy-Wolff Property in simply connected domains using $H$-limits and automorphisms, expanding understanding beyond visibility conditions.

Findings

Denjoy-Wolff Property characterized by $H$-limits at boundary points.

Existence of domains with the property that are not visible, disproving necessity of visibility.

Equivalence established between the property and automorphisms without fixed points.

Abstract

We characterize the simply connected domains $\Omega\subsetneq\mathbb{C}$ that exhibit the Denjoy-Wolff Property, meaning that every holomorphic self-map of $\Omega$ without fixed points has a Denjoy-Wolff point. We demonstrate that this property holds if and only if every automorphism of $\Omega$ without fixed points in $\Omega$ 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 exist 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.