Spiders' webs in the Eremenko-Lyubich class
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909606751502336 |
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| author | Rempe, Lasse |
| author_facet | Rempe, Lasse |
| contents | Consider the entire function $f(z)=\cosh(z)$. We show that the escaping set of this function - that is, the set of points whose orbits tend to infinity under iteration - has a structure known as a "spider's web". This disproves a conjecture of Sixsmith from 2020. In fact, we show that the "fast escaping set", i.e. the set of points whose orbits tend to infinity at an iterated exponential rate, is a spider's web. This answers a question of Rippon and Stallard from 2012. We also discuss a wider class of functions to which our results apply, and state some open questions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20998 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spiders' webs in the Eremenko-Lyubich class Rempe, Lasse Dynamical Systems Complex Variables Primary 37F10, secondary 30D05 Consider the entire function $f(z)=\cosh(z)$. We show that the escaping set of this function - that is, the set of points whose orbits tend to infinity under iteration - has a structure known as a "spider's web". This disproves a conjecture of Sixsmith from 2020. In fact, we show that the "fast escaping set", i.e. the set of points whose orbits tend to infinity at an iterated exponential rate, is a spider's web. This answers a question of Rippon and Stallard from 2012. We also discuss a wider class of functions to which our results apply, and state some open questions. |
| title | Spiders' webs in the Eremenko-Lyubich class |
| topic | Dynamical Systems Complex Variables Primary 37F10, secondary 30D05 |
| url | https://arxiv.org/abs/2410.20998 |