Topological Structure of Fractal Squares

Given an integer $n\geq 2$ and a digit set ${\mathcal D}\subsetneq \{0,1,...,n-1\}^2$, there is a self-similar set satisfying the set equation: $F=(F+{\mathcal D})/n$. We call such $F$ a fractal square. By studying a torus-like structure of $H= F+ {\mathbb Z}^2$, we classify $F$ into three types according to their topological properties. We also provide some simple criteria for such classification.