Structures of moduli spaces of generalized Cantor sets
Hiroshige Shiga
TL;DR
This work analyzes moduli spaces $\mathcal M(\omega)$ of generalized Cantor sets $E(\omega)$ under quasiconformal equivalence. It proves the moduli spaces are measurable ($F_{\sigma}$) subsets of $\Omega=(0,1)^{\mathbb N}$ and establishes a necessary condition relating sequences $\omega$ and $\omega'$ for qc-equivalence, using canonical pants decompositions and extremal length techniques. The authors show there are uncountably many moduli spaces and that, up to the standard product measure, the volume of almost all moduli spaces is zero, with at most one exceptional moduli space possibly having positive volume due to shift-ergodicity. They also construct explicit uncountable families by varying $\omega$ (via $q_n(\alpha)=1-e^{-n^{\alpha}}$) to demonstrate the richness of the moduli space structure, contrasting with finiteness results known for Kleinian groups. The methods combine hyperbolic geometry, extremal length, kernel convergence, and ergodic theory to reveal a fine moduli-theoretic landscape for Cantor-type sets.
Abstract
For each $ω\in (0, 1)^{\mathbb N}$, we may construct a Cantor set $E(ω)\subset [0, 1]$ called a generalized Cantor set for $ω$. We study the moduli space of $ω$ denoted by $\mathcal M(ω)\subset (0, 1)^{\mathbb N}$. It is the set of $ω'$ so that $E(ω')$ is quasiconformally equivalent to $E(ω)$. In this paper, we show that the set $\mathcal M(ω)$ is measurable in $(0, 1)^{\mathbb N}$ and we give a necessary condition for $ω'$ to belong to $\mathcal M(ω)$. 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.