Topological Structure of Fractal Squares

Abstract

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

17 pages, 12 figures

References in corpus (2)

• Lipschitz equivalence of self-similar sets and hyperbolic boundaries
• Connectedness of planar self-affine sets associated with non-consecutive collinear digit sets

Cited by in corpus (4)

• Lipschitz equivalence of self-similar sets and hyperbolic boundaries
• Mixed labyrinth fractals
• Fractal squares with finitely many connected components
• Connectedness of self-affine sets with product digit sets