Topological structure of fractal squares

Abstract

AbstractGiven an integer n ≥ 2 and a digit set ⊊ {0,1,. . .,n − 1}2, there is a self-similar set F ⊂ ℝ2 satisfying the set equation: F=(F+)/n. We call such F a fractal square. By studying a periodic extension H= F + ℤ2, we classify F into three types according to their topological properties. We also provide some simple criteria for such classification.