On a class of self-similar sets which contain finitely many common points
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866917683385073664 |
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| author | Jiang, Kan Kong, Derong Li, Wenxia Wang, Zhiqiang |
| author_facet | Jiang, Kan Kong, Derong Li, Wenxia Wang, Zhiqiang |
| contents | For $λ\in(0,1/2]$ let $K_λ\subset\mathbb{R}$ be a self-similar set generated by the iterated function system $\{λx, λx+1-λ\}$. Given $x\in(0,1/2)$, let $Λ(x)$ be the set of $λ\in(0,1/2]$ such that $x\in K_λ$. In this paper we show that $Λ(x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any $y_1,\ldots, y_p\in(0,1/2)$ there exists a full Hausdorff dimensional set of $λ\in(0,1/2]$ such that $y_1,\ldots, y_p \in K_λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_10014 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On a class of self-similar sets which contain finitely many common points Jiang, Kan Kong, Derong Li, Wenxia Wang, Zhiqiang Dynamical Systems Classical Analysis and ODEs Primary: 28A78, Secondary: 28A80, 37B10 For $λ\in(0,1/2]$ let $K_λ\subset\mathbb{R}$ be a self-similar set generated by the iterated function system $\{λx, λx+1-λ\}$. Given $x\in(0,1/2)$, let $Λ(x)$ be the set of $λ\in(0,1/2]$ such that $x\in K_λ$. In this paper we show that $Λ(x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any $y_1,\ldots, y_p\in(0,1/2)$ there exists a full Hausdorff dimensional set of $λ\in(0,1/2]$ such that $y_1,\ldots, y_p \in K_λ$. |
| title | On a class of self-similar sets which contain finitely many common points |
| topic | Dynamical Systems Classical Analysis and ODEs Primary: 28A78, Secondary: 28A80, 37B10 |
| url | https://arxiv.org/abs/2109.10014 |