Topological Structure of Fractal Squares

TL;DR

This paper classifies fractal squares, self-similar sets generated by digit sets, into three topological types using periodic extensions and provides criteria for this classification.

Contribution

It introduces a classification scheme for fractal squares based on their topological properties and offers simple criteria for determining their type.

Findings

Fractal squares can be categorized into three topological types.

Periodic extension helps in classifying the topological structure.

Criteria are provided for easy classification of fractal squares.

Abstract

Given an integer $n\geq 2$ and a digit set ${\mathcal D}\subsetneq {0,1,...,n-1}^2$, there is a self-similar set $F \subset {\Bbb R}^2$ satisfying the set equation: $F=(F+{\mathcal D})/n$. We call such $F$ a fractal square. By studying a periodic extension $H= F+ {\mathbb Z}^2$, we classify $F$ into three types according to their topological properties. We also provide some simple criteria for such classification.