
TL;DR
This paper constructs a self-similar genus 2 Cantor set in three-dimensional space, demonstrating the first example with uniform local genus 2 and linking it to a new class of quasiregular mappings with such Julia sets.
Contribution
It introduces the first example of a genus 2 Cantor set with uniform local genus and connects it to a novel quasiregular mapping with this Julia set.
Findings
Constructed a self-similar genus 2 Cantor set in .
Established the first example with uniform local genus 2.
Linked the Cantor set to a new quasiregular mapping with this Julia set.
Abstract
We construct a geometrically self-similar Cantor set of genus in . This construction is the first for which the local genus is shown to be at every point of . As an application, we construct, also for the first time, a uniformly quasiregular mapping for which the Julia set is a genus Cantor set.
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