The Denjoy-Wolff Theorem in simply connected domains
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914951761756160 |
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| author | Benini, Anna Miriam Bracci, Filippo |
| author_facet | Benini, Anna Miriam Bracci, Filippo |
| contents | We characterize the simply connected domains $Ω\subsetneq\mathbb{C}$ that exhibit the Denjoy-Wolff Property, meaning that every holomorphic self-map of $Ω$ without fixed points has a Denjoy-Wolff point. We demonstrate that this property holds if and only if every automorphism of $Ω$ without fixed points in $Ω$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11722 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Denjoy-Wolff Theorem in simply connected domains Benini, Anna Miriam Bracci, Filippo Complex Variables We characterize the simply connected domains $Ω\subsetneq\mathbb{C}$ that exhibit the Denjoy-Wolff Property, meaning that every holomorphic self-map of $Ω$ without fixed points has a Denjoy-Wolff point. We demonstrate that this property holds if and only if every automorphism of $Ω$ without fixed points in $Ω$ 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. |
| title | The Denjoy-Wolff Theorem in simply connected domains |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2409.11722 |