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 $\omega'$ so that $E(\omega')$ 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 $\omega'$ 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.