Self-similarity of unions of self-similar sets and their translations

TL;DR

This paper characterizes when unions of translated self-similar sets remain self-similar, using graph cycles, and identifies cases where self-similarity does not hold after translation.

Contribution

It provides a complete characterization of translation vectors preserving self-similarity in unions of self-similar sets, extending previous results and identifying non-preserving cases.

Findings

Characterization of translation vectors leading to self-similar unions.

Use of directed graphs to determine self-similarity conditions.

Identification of self-similar sets whose unions with translations are not self-similar.

Abstract

In this paper, we explore the self-similarity of unions of self-similar sets and their translations. For $N \in \mathbb{N}$ and $0< \beta < 1/(N+1)$, let $\Gamma$ be the self-similar set generated by the IFS \[ \Big\{ \phi_i(x)=\beta x + i \frac{1-\beta}{N}: i=0,1,\ldots, N \Big\}. \] We provide a complete characterization of translation vectors $\boldsymbol{t} =(t_0,t_1, \ldots, t_m) \in \mathbb{R}^{m+1}$ with $0=t_0 < t_1 < \cdots < t_m$ for which the union $\bigcup_{j=0}^m (\Gamma+t_j)$ is a self-similar set, by determining the existence of cycles in associated directed graphs. This extends the result of [Derong Kong, Wenxia Li, Zhiqiang Wang, Yuanyuan Yao, Yunxiu Zhang. On the union of homogeneous symmetric Cantor set with its translations. Math. Z., 2024]. Additionally, we present two types of self-similar sets for which the union with their translations cannot be self-similar.