On the moduli space of the standard Cantor set

TL;DR

This paper characterizes when a generalized Cantor set, defined by an infinite sequence, is quasiconformally equivalent to the standard middle-third Cantor set, providing a complete criterion based on the sequence.

Contribution

It establishes a necessary and sufficient condition for quasiconformal equivalence between generalized and standard Cantor sets based on their defining sequences.

Findings

Provides a complete criterion for quasiconformal equivalence

Characterizes the moduli space of generalized Cantor sets

Links sequence properties to geometric equivalence

Abstract

We consider a generalized Cantor set $E(\omega)$ for an infinite sequence $\omega=(q_n)_{n=1}^{\infty}$ of positive numbers with $0<q_n<1$, and examine the quasiconformal equivalence to the standard middle one-third Cantor set $E(\omega_0)$. We may give a necessary and sufficient condition for $E(\omega)$ to be quasiconformally equivalent to $E(\omega_0)$ in terms of $\omega$.