Self-similarity of unions of self-similar sets and their translations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915980711559168 |
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| author | Wang, Zhiqiang |
| author_facet | Wang, Zhiqiang |
| contents | In this paper, we explore the self-similarity of unions of self-similar sets and their translations. For $N \in \mathbb{N}$ and $0< β< 1/(N+1)$, let $Γ$ 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 $\boldsymbol{t} =(t_0,t_1, \ldots, t_m) \in \mathbb{R}^{m+1}$ with $0=t_0 < t_1 < \cdots < t_m$ for which the union $\bigcup_{j=0}^m (Γ+t_j)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_01824 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Self-similarity of unions of self-similar sets and their translations Wang, Zhiqiang Dynamical Systems Classical Analysis and ODEs 28A80 In this paper, we explore the self-similarity of unions of self-similar sets and their translations. For $N \in \mathbb{N}$ and $0< β< 1/(N+1)$, let $Γ$ 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 $\boldsymbol{t} =(t_0,t_1, \ldots, t_m) \in \mathbb{R}^{m+1}$ with $0=t_0 < t_1 < \cdots < t_m$ for which the union $\bigcup_{j=0}^m (Γ+t_j)$ 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. |
| title | Self-similarity of unions of self-similar sets and their translations |
| topic | Dynamical Systems Classical Analysis and ODEs 28A80 |
| url | https://arxiv.org/abs/2605.01824 |