Phragmén-Lindelöf Principles and Julia Limiting Directions of Quasiregular Mappings
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911871161860096 |
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| author | Fletcher, Alastair N. Steranka, Julie M. |
| author_facet | Fletcher, Alastair N. Steranka, Julie M. |
| contents | We show that the set of Julia limiting directions of a transcendental-type $K$-quasiregular mapping $f:\mathbb{R}^n\to \mathbb{R}^n$ must contain a component of a certain size, depending on the dimension $n$, the maximal dilatation $K$, and the order of growth of $f$. In particular, we show that if the order of growth is small enough, then every direction is a Julia limiting direction. We also show that if every component of the set of Julia limiting directions is a point, then $f$ has infinite order. The main tool in proving these results is a new version of a Phragmén-Lindelöf principle for sub-$F$-extremals in sectors, where we allow for boundary growth of the form $O( \log |x| )$ instead of the previously considered $O(1)$ bound. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_17053 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Phragmén-Lindelöf Principles and Julia Limiting Directions of Quasiregular Mappings Fletcher, Alastair N. Steranka, Julie M. Dynamical Systems Complex Variables 37F31 (Primary) 30C65, 31C45 (Secondary) We show that the set of Julia limiting directions of a transcendental-type $K$-quasiregular mapping $f:\mathbb{R}^n\to \mathbb{R}^n$ must contain a component of a certain size, depending on the dimension $n$, the maximal dilatation $K$, and the order of growth of $f$. In particular, we show that if the order of growth is small enough, then every direction is a Julia limiting direction. We also show that if every component of the set of Julia limiting directions is a point, then $f$ has infinite order. The main tool in proving these results is a new version of a Phragmén-Lindelöf principle for sub-$F$-extremals in sectors, where we allow for boundary growth of the form $O( \log |x| )$ instead of the previously considered $O(1)$ bound. |
| title | Phragmén-Lindelöf Principles and Julia Limiting Directions of Quasiregular Mappings |
| topic | Dynamical Systems Complex Variables 37F31 (Primary) 30C65, 31C45 (Secondary) |
| url | https://arxiv.org/abs/2303.17053 |