The connectedness of Sierpiński sponges with rotational and reflectional components and associated graph-directed systems
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arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916557042483200 |
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| author | Ruan, Huo-Jun Xiao, Jian-Ci |
| author_facet | Ruan, Huo-Jun Xiao, Jian-Ci |
| contents | We provide two methods to characterize the connectedness of all $d$-dimensional generalized Sierpiński sponges whose corresponding IFSs are allowed to have rotational and reflectional components. Our approach is to reduce it to an intersection problem between the coordinates of graph-directed attractors. More precisely, let $(K_1,\ldots,K_n)$ be a Cantor-type graph-directed attractor in $\mathbb{R}^d$. By creating an auxiliary graph, we provide an effective criterion for whether $K_i\cap K_j$ is empty for every pair of $1\leq i,j\leq n$. Moreover, the emptiness can be checked by examining only a finite number of geometric approximations of the attractor. The approach is also applicable to more general graph-directed systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_09219 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The connectedness of Sierpiński sponges with rotational and reflectional components and associated graph-directed systems Ruan, Huo-Jun Xiao, Jian-Ci General Topology Primary 28A80, Secondary 54A05 We provide two methods to characterize the connectedness of all $d$-dimensional generalized Sierpiński sponges whose corresponding IFSs are allowed to have rotational and reflectional components. Our approach is to reduce it to an intersection problem between the coordinates of graph-directed attractors. More precisely, let $(K_1,\ldots,K_n)$ be a Cantor-type graph-directed attractor in $\mathbb{R}^d$. By creating an auxiliary graph, we provide an effective criterion for whether $K_i\cap K_j$ is empty for every pair of $1\leq i,j\leq n$. Moreover, the emptiness can be checked by examining only a finite number of geometric approximations of the attractor. The approach is also applicable to more general graph-directed systems. |
| title | The connectedness of Sierpiński sponges with rotational and reflectional components and associated graph-directed systems |
| topic | General Topology Primary 28A80, Secondary 54A05 |
| url | https://arxiv.org/abs/2204.09219 |