On the classification of fractal squares

TL;DR

This paper advances the understanding of fractal squares by providing criteria for total disconnectedness and exploring Lipschitz classification for the case n=3, extending previous topological classifications.

Contribution

It introduces simple criteria for total disconnectedness of fractal squares and investigates Lipschitz classification for the non-totally disconnected case when n=3.

Findings

Criteria for total disconnectedness of fractal squares

Lipschitz classification results for n=3 case

Extension of topological classification to non-totally disconnected sets

Abstract

In \cite{LaLuRa13}, the authors completely classified the topological structure of so called {\it fractal square} $F$ defined by $F=(F+{\mathcal D})/n$, where ${\mathcal{D}}\subsetneq\{0,1,\dots,n-1\}^2, n\ge 2$. In this paper, we further provide simple criteria for the $F$ to be totally disconnected, then we discuss the Lipschitz classification of $F$ in the case $n=3$, which is an attempt to consider non-totally disconnected sets.