Fractal squares with finitely many connected components

TL;DR

This paper introduces a method to determine when a disconnected fractal square has finitely many connected components by using graph structures, and proves that such fractal squares contain either finitely or uncountably many components.

Contribution

It provides a novel graph-based characterization method for connectedness in fractal squares and classifies their number of components as finite or uncountably infinite.

Findings

A complete characterization of when fractal squares have finitely many components.

Proof that fractal squares have either finitely or uncountably many components.

Examples of fractal squares with exactly m connected components for m ≥ 2.

Abstract

In this paper, we present an effective method to characterize completely when a disconnected fractal square has only finitely many connected components. Our method is to establish some graph structures on fractal squares to reveal the evolution of the connectedness during their geometric iterated construction. We also prove that every fractal square contains either finitely or uncountably many connected components. A few examples, including the construction of fractal squares with exactly $m\geqslant 2$ connected components, are added in addition.