Boundary dynamics in unbounded Fatou components
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914833985699840 |
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| author | Jové, Anna Fagella, Núria |
| author_facet | Jové, Anna Fagella, Núria |
| contents | We study the behaviour of a transcendental entire map $ f\colon \mathbb{C}\to\mathbb{C} $ on an unbounded invariant Fatou component $ U $, assuming that infinity is accessible from $ U $. It is well-known that $ U $ is simply connected. Hence, by means of a Riemann map $ φ\colon\mathbb{D}\to U $ and the associated inner function, the boundary of $ U $ is described topologically in terms of the disjoint union of clusters sets, each of them consisting of one or two connected components in $ \mathbb{C} $.
Moreover, under more precise assumptions on the distribution of singular values, it is proven that periodic and escaping boundary points are dense in $ \partial U $, being all periodic boundary points accessible from $ U $. Finally, under the same conditions, the set of singularities of $ g $ is shown to have zero Lebesgue measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_11384 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Boundary dynamics in unbounded Fatou components Jové, Anna Fagella, Núria Dynamical Systems We study the behaviour of a transcendental entire map $ f\colon \mathbb{C}\to\mathbb{C} $ on an unbounded invariant Fatou component $ U $, assuming that infinity is accessible from $ U $. It is well-known that $ U $ is simply connected. Hence, by means of a Riemann map $ φ\colon\mathbb{D}\to U $ and the associated inner function, the boundary of $ U $ is described topologically in terms of the disjoint union of clusters sets, each of them consisting of one or two connected components in $ \mathbb{C} $. Moreover, under more precise assumptions on the distribution of singular values, it is proven that periodic and escaping boundary points are dense in $ \partial U $, being all periodic boundary points accessible from $ U $. Finally, under the same conditions, the set of singularities of $ g $ is shown to have zero Lebesgue measure. |
| title | Boundary dynamics in unbounded Fatou components |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2307.11384 |