Self-similar subsets of the Cantor set

TL;DR

This paper characterizes all self-similar subsets of the middle-third Cantor set, providing criteria for their classification based on the contraction ratios of their generating iterated function systems.

Contribution

It offers a complete classification of self-similar subsets of the Cantor set, including explicit criteria for their generating functions and contraction ratios.

Findings

Self-similar subsets with more than one point have generating IFS with ratios b1 3^{-n}.

A simple criterion characterizes subsets with equal contraction ratios.

Provides a complete classification of such subsets.

Abstract

In this paper, we study the following question raised by Mattila in 1998: what are the self-similar subsets of the middle-third Cantor set $\C$? We give criteria for a complete classification of all such subsets. We show that for any self-similar subset $\F$ of $\C$ containing more than one point every linear generating IFS of $\F$ must consist of similitudes with contraction ratios $\pm 3^{-n}$, $n\in \N$. In particular, a simple criterion is formulated to characterize self-similar subsets of $\C$ with equal contraction ratio in modulus.