Structures of moduli spaces of generalized Cantor sets

TL;DR

This paper investigates the structure and measure-theoretic properties of the moduli spaces of generalized Cantor sets, revealing their measurability, uncountability, and that most have zero volume under the standard measure.

Contribution

It establishes the measurability of moduli spaces, provides a necessary condition for membership, and demonstrates the uncountability and measure-zero property of these spaces.

Findings

The moduli space set is measurable.

There are uncountably many moduli spaces.

Most moduli spaces have zero volume measure.

Abstract

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.