Topological properties of self-similar fractals with one parameter

Abstract

In this paper, we study two classes of planar self-similar fractals $T_{\epsilon }$ with a shifting parameter $\epsilon$. The first one is a class of self-similar tiles by shifting $x$-coordinates of some digits. We give a detailed discussion on the disk-likeness ({\it i.e., the property of being a topological disk}) in terms of $\epsilon$. We also prove that $T_{\epsilon }$ determines a quasi-periodic tiling if and only if $\epsilon$ is rational. The second one is a class of self-similar sets by shifting diagonal digits. We give a necessary and sufficient condition for $T_{\epsilon }$ to be connected.

arXiv comment17 pages, 12figures