Lipschitz equivalence of a pair of fractal squares

Abstract

The fractal square $F$ is defined by $F=\frac{1}{n}(F+\mathcal{D})$, where $\mathcal{D}=\{d_{1},d_{2},\ldots,d_{m}\}\subseteq\{0,1,\ldots,n-1\}^{2},~$ $n\geq~2$. In this paper, we study the structure of two non-totally disconnected fractal squares in the case $m=6$, $n=3$ and construct a map between them. Using finite state automaton, we prove that this map is a bi-Lipschitz map.