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

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 \[ \Big\{ ϕ_i(x)=βx + i \frac{1-β}{N}: i=0,1,\ldots, N \Big\}. \] We provide a complete characterization of translation vectors $\mathbf{t}=(t_0,t_1,\dots ,t_m)\in {\mathbb{R}}^{m+1}$ 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.

23 pages