Uniform domains and moduli spaces of generalized Cantor sets

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 $\omega$.