Topological properties of self-similar fractals with one parameter

TL;DR

This paper investigates the topological and connectivity properties of two classes of planar self-similar fractals with a shifting parameter, revealing conditions for disk-likeness, tiling, and connectivity.

Contribution

It provides a detailed analysis of how the parameter affects topological properties and establishes criteria for disk-likeness, tiling, and connectivity of the fractals.

Findings

Disk-likeness depends on the parameter psilon.

Quasi-periodic tilings occur if and only if psilon is rational.

Connectivity conditions are characterized by specific parameter values.

Abstract

In this paper, we study two classes of planar self-similar fractals $T_\varepsilon$ with a shifting parameter $\varepsilon$. 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 $\varepsilon$. We also prove that $T_\varepsilon$ determines a quasi-periodic tiling if and only if $\varepsilon$ is rational. The second one is a class of self-similar sets by shifting diagonal digits. We give a necessary and sufficient condition for $T_\varepsilon$ to be connected.