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- W4306295079 abstract "The main aim of this paper is to give topological obstructions to Cantor sets in $mathbb{R}^3$ being Julia sets of uniformly quasiregular mappings. Our main tool is the genus of a Cantor set. We give a new construction of a genus $g$ Cantor set, the first for which the local genus is $g$ at every point, and then show that this Cantor set can be realized as the Julia set of a uniformly quasiregular mapping. These are the first such Cantor Julia sets constructed for $ggeq 3$. We then turn to our dynamical applications and show that every Cantor Julia set of a hyperbolic uniformly quasiregular map has a finite genus $g$; that a given local genus in a Cantor Julia set must occur on a dense subset of the Julia set; and that there do exist Cantor Julia sets where the local genus is non-constant." @default.
- W4306295079 created "2022-10-15" @default.
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- W4306295079 date "2022-10-12" @default.
- W4306295079 modified "2023-09-27" @default.
- W4306295079 title "Genus $g$ Cantor sets and germane Julia sets" @default.
- W4306295079 doi "https://doi.org/10.48550/arxiv.2210.06619" @default.
- W4306295079 hasPublicationYear "2022" @default.
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