A Classification of Fractal Squares

Abstract

Let $\hat{I}{»}_K:{\mathbb{R}}^2\to \{0,1,\dots \}\cup \{\infty \}$ 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, $\hat{I}{»}_K(K)=\{0\}$. In this article we show that every fractal square $K$ satisfies $\hat{I}{»}_K(K)\subset \{0,1\}$ and find criterions to classify when $\hat{I}{»}_K(K)$ equals $\{0\}$, $\{1\}$ or $\{0,1\}$. Here for any integer $N\ge 2$ and any set $\Dc=\left\{(i,j): 0\le i,j\le N-1\right\}$ with cardinality $\ge 2$, 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 $K={\bigcap }_n{K}^{(n)}$ is called a fractal square.

23 pages, 13 figures