Uniform domains and moduli spaces of generalized Cantor sets

TL;DR

This paper characterizes when complements of generalized Cantor sets are uniform domains and explores the structure and measure of their moduli spaces, revealing conditions for conformal equivalence and measure vanishing.

Contribution

It provides a necessary and sufficient condition for the complement of a generalized Cantor set to be a uniform domain and analyzes the measure of their moduli spaces.

Findings

Characterization of uniform domains among complements of generalized Cantor sets.

Conditions for generalized Cantor sets to belong to specific moduli spaces.

The volume of the moduli space can vanish under certain conditions.

Abstract

We consider a generalized Cantor set $E(\omega)$ for an infinite sequence $\omega=(q_n)_{n=1}^{\infty}\in (0, 1)^{\mathbb N}$, and consider the moduli space $M(\omega)$ for $\omega$ which are the set of $\omega'$ for which $E(\omega')$ is conformally equivalent to $E(\omega)$. In this paper, we may give a necessary and sufficient condition for $D(\omega):=\mathbb C\setminus E(\omega)$ to be a uniform domain. As a byproduct, we give a condition for $E(\omega)$ to belong to $M(\omega_0)$, the moduli space of the standard middle one-third Cantor set. We also show that the volume of the moduli space $M(\omega)$ with respect to the standard product measure on $(0, 1)^{\mathbb N}$ vanishes under a certain condition for $\omega$.