Structures of moduli spaces of generalized Cantor sets

For each $\omega \in (0,1{)}^{\mathbb{N}}$, we may construct a Cantor set $E(\omega )\subset [0,1]$ called a generalized Cantor set for $\omega$. We study the moduli space of $\omega$ denoted by $\mathcal{M}(\omega )\subset (0,1{)}^{\mathbb{N}}$. It is the set of ${\omega }^{\mathrm{\prime }}$ so that $E({\omega }^{\mathrm{\prime }})$ is quasiconformally equivalent to $E(\omega )$. In this paper, we show that the set $\mathcal{M}(\omega )$ is measurable in $(0,1{)}^{\mathbb{N}}$ and we give a necessary condition for ${\omega }^{\mathrm{\prime }}$ to belong to $\mathcal{M}(\omega )$. 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.