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Genus 2 Cantor sets

Alastair Fletcher and Daniel Stoertz

Vol. 321 (2022), No. 2, 283–307
Abstract

We construct a geometrically self-similar Cantor set X of genus 2 in 3. This construction is the first for which the local genus is shown to be 2 at every point of X. As an application, we construct, also for the first time, a uniformly quasiregular mapping f : 3 3 for which the Julia set J(f) is a genus 2 Cantor set.

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Keywords
Cantor sets, Julia sets, uniformly quasiregular mappings
Mathematical Subject Classification
Primary: 54C50
Secondary: 30C65, 37F10, 37F50
Milestones
Received: 25 August 2021
Revised: 31 October 2022
Accepted: 5 November 2022
Published: 21 March 2023
Authors
Alastair Fletcher
Department of Mathematical Sciences
Northern Illinois University
Dekalb, IL
United States
Daniel Stoertz
Department of Mathematics, Statistics, and Computer Science
St. Olaf College
Northfield, MN
United States