A Classification of Fractal Squares

Let ${\lambda }_K:{\text{bbR}}^2\to \{0,1,\dots \}\cup \{\mathrm{\infty }\}$ 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 $N\ge 2$ and any set $\text{Dc}=\{(i,j):0\le i,j\le N-1\}$ with cardinality $\ge 2$, if we set $K^{(0)}=[0,1{]}^2$ and ${\displaystyle K^{(n)}=\{\frac{x+d}{N}:x\in K^{(n-1)},d\in \text{Dc}\}(n\ge 1)}$ then $K={\bigcap }_n{K}^{(n)}$ is called a fractal square.