Tohoku Mathematical Journal
2026
March
SECOND SERIES VOL. 78, NO. 1

Tohoku Math. J.
78 (2026), 85-100

Title UNIFORM DOMAINS AND MODULI SPACES OF GENERALIZED CANTOR SETS

Author Hiroshige Shiga

(Received September 11, 2023, revised April 1, 2024)
Abstract. We consider a generalized Cantor set $E(\omega)$ for an infinite sequence $\omega=(q_n)_{n=1}^{\infty}\in (0, 1)^{\mathbb N}$, and consider the moduli space $M(\omega)$ for $\omega$ which are the set of $\omega'$ for which $E(\omega')$ 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 $\omega$.

Mathematics Subject Classification. Primary 30C62; Secondary 30F45.
Key words and phrases. Cantor set, Quasiconformal mapping.

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