The Denjoy-Wolff Theorem in simply connected domains

Anna Miriam Benini, Filippo Bracci
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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.
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简单相连域中的登乔伊-沃尔夫定理
我们描述了表现出 Denjoy-Wolff 特性的简单连接域 $\Omega\subsetneq\mathbb{C}$ 的特征,这意味着 $\Omega$ 的每个无定点的全形自映射都有一个 Denjoy-Wolff 点。我们证明,当且仅当 $\Omega$ 中没有定点的 $\Omega$ 的每个自变都有一个 Denjoy-Wolff 点时,这个性质才成立。此外,我们还证明了登喜-沃尔夫性质等同于与域相关的黎曼图在每个边界点上存在我们称之为"$H$极限 "的条件。$H$极限条件强于非切线极限的存在,但弱于无限制极限。作为我们工作的附加结果,我们证明存在有界简单相连域,其中登乔伊-沃尔夫性质成立,但在巴拉利和齐默尔的意义上不可见。由于可见性是登乔伊-沃尔夫性质的充分条件,这就证明了在一般情况下,可见性并不是必要条件。
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