Topological Structure of Fractal Squares
Ka-Sing Lau, Jun Jason Luo, Hui Rao

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 and a digit set , there is a self-similar set satisfying the set equation: . We call such a fractal square. By studying a periodic extension , we classify into three types according to their topological properties. We also provide some simple criteria for such classification.
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