Structures of moduli spaces of generalized Cantor sets

2025/12/16 by Hiroshige Shiga, Shiga, Hiroshige
Mathematics · #Analytic and geometric function theory #Mathematical Dynamics and Fractals #Advanced Topology and Set Theory

paper · pdf · doi:10.48550/arxiv.2512.13990

Abstract

For each ω∈ (0, 1)^{\mathbb N}, we may construct a Cantor set E(ω)⊂ [0, 1] called a generalized Cantor set for ω. We study the moduli space of ω denoted by \mathcal M(ω)⊂ (0, 1)^{\mathbb N}. It is the set of ω' so that E(ω') is quasiconformally equivalent to E(ω). In this paper, we show that the set \mathcal M(ω) is measurable in (0, 1)^{\mathbb N} and we give a necessary condition for ω' to belong to \mathcal M(ω). By using this condition, we show that there are uncountably many moduli spaces in (0, 1)^{\mathbb N}. We also show that except for at most one moduli space, the volume of the moduli space with respect to the standard product measure of (0, 1)^{\mathbb N} vanishes.

Citations

• Uniform domains and moduli spaces of generalized Cantor sets
• On quasiconformal equivalence of Schottky regions
• On the quasiconformal equivalence of dynamical Cantor sets
• The quasiconformal equivalence of Riemann surfaces and the universal Schottky space

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• Uniform domains and moduli spaces of generalized Cantor sets
• On the moduli space of the standard Cantor set
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• Conformal dimension: Cantor sets and moduli
• Diophantine sets in general are Cantor sets