On the failure of the Denjoy-Wolff Theorem in convex domains
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911447014965248 |
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| author | Bracci, Filippo Ökten, Ahmed Yekta |
| author_facet | Bracci, Filippo Ökten, Ahmed Yekta |
| contents | In this note, we construct examples of bounded smooth convex domains with no non-trivial analytic discs on the boundary which possess a holomorphic self-map without fixed points so that the iterates do not converge to a point (that is, the Denjoy-Wolff theorem does not hold). We also show that, in the case of bounded convex domains with $C^{1+\varepsilon}$-smooth boundary which have non-trivial analytic discs on the boundary, the cluster set of the orbits of holomorphic self-maps without fixed points can be equal to the principal part of any prime end of any planar bounded simply connected domain. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_10298 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the failure of the Denjoy-Wolff Theorem in convex domains Bracci, Filippo Ökten, Ahmed Yekta Complex Variables Dynamical Systems 30C35, 30D05, 30D40, 32F45 In this note, we construct examples of bounded smooth convex domains with no non-trivial analytic discs on the boundary which possess a holomorphic self-map without fixed points so that the iterates do not converge to a point (that is, the Denjoy-Wolff theorem does not hold). We also show that, in the case of bounded convex domains with $C^{1+\varepsilon}$-smooth boundary which have non-trivial analytic discs on the boundary, the cluster set of the orbits of holomorphic self-maps without fixed points can be equal to the principal part of any prime end of any planar bounded simply connected domain. |
| title | On the failure of the Denjoy-Wolff Theorem in convex domains |
| topic | Complex Variables Dynamical Systems 30C35, 30D05, 30D40, 32F45 |
| url | https://arxiv.org/abs/2507.10298 |