Uniform domains and moduli spaces of generalized Cantor sets

We consider a generalized Cantor set $E(\omega )$ for an infinite sequence $\omega =(q_n{)}_{n=1}^{\mathrm{\infty }}\in (0,1{)}^{\mathbb{N}}$, and consider the moduli space $M(\omega )$ for $\omega$ which are the set of ${\omega }^{\mathrm{\prime }}$ for which $E({\omega }^{\mathrm{\prime }})$ 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 $\omega$.