A Classification of Fractal Squares

TL;DR

This paper classifies fractal squares based on their lambda function, showing it takes values in {0,1} and providing criteria for when it equals each, thus linking fractal structure to local connectivity properties.

Contribution

It introduces a classification scheme for fractal squares using the lambda function and establishes criteria for their lambda values, advancing understanding of their topological properties.

Findings

Lambda function values are contained in {0,1} for fractal squares.

Criteria for when lambda equals 0, 1, or both are established.

Connection between lambda function values and local connectivity of fractal squares.

Abstract

Let $\lambda_K:\bbR^2\rightarrow\{0,1,\ldots\}\cup\{\infty\}$ be the lambda function of a planar comapctum $K$, as defined in MR4488162. It is known that a planar continuum is locally connected if and only if its lambda function vanishes everywhere, or equivalently, $\lambda_K(K)=\{0\}$. In this article we show that every fractal square $K$ satisfies $\lambda_K(K)\subset\{0,1\}$ and find criterions to classify when $\lambda_K(K)$ equals $\{0\}$, $\{1\}$ or $\{0,1\}$. Here for any integer $N\ge2$ and any set $\Dc=\left\{(i,j): 0\le i,j\le N-1\right\}$ with cardinality $\ge2$, if we set $K^{(0)}=[0,1]^2$ and $\displaystyle K^{(n)}=\left\{\frac{x+d}{N}: x\in K^{(n-1)}, d\in\Dc\right\}(n\ge1)$ then $K=\bigcap_nK^{(n)}$ is called a fractal square.