Uniform domains and moduli spaces of generalized Cantor sets

2023/09/04 by Hiroshige Shiga, Shiga, Hiroshige · 1 citation
Mathematics · #30C62 #Advanced Topology and Set Theory #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2309.01303

openalex publication_date 2023/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a generalized Cantor set E(ω) for an infinite sequence ω=(q_n)_n=1^∞∈ (0, 1)^{\mathbb N}, and consider the moduli space M(ω) for ω which are the set of ω' for which E(ω') is conformally equivalent to E(ω). In this paper, we may give a necessary and sufficient condition for D(ω):=\mathbb C∖ E(ω) to be a uniform domain. As a byproduct, we give a condition for E(ω) to belong to M(ω₀), the moduli space of the standard middle one-third Cantor set. We also show that the volume of the moduli space M(ω) with respect to the standard product measure on (0, 1)^{\mathbb N} vanishes under a certain condition for ω.

Cited by

• Structures of moduli spaces of generalized Cantor sets

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