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| Natura: | Preprint |
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2021
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| Accesso online: | https://arxiv.org/abs/2107.12076 |
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| _version_ | 1866910447018442752 |
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| author | Thuswaldner, Jörg M. Zhang, Shu-Qin |
| author_facet | Thuswaldner, Jörg M. Zhang, Shu-Qin |
| contents | Let $M$ be a $3\times 3$ integer matrix which is expanding in the sense that each of its eigenvalues is greater than $1$ in modulus and let $\mathcal{D} \subset \mathbb{Z}^3$ be a digit set containing $|\det M|$ elements. Then the unique nonempty compact set $T=T(M,\mathcal{D})$ defined by the set equation $MT=T+\mathcal{D}$ is called an integral self-affine tile if its interior is nonempty. If $\mathcal{D}$ is of the form $\mathcal{D}=\{0,v,\ldots, (|\det M|-1)v\}$ we say that $T$ has a collinear digit set. The present paper is devoted to the topology of integral self-affine tiles with collinear digit sets. In particular, we prove that a large class of these tiles is homeomorphic to a closed $3$-dimensional ball. Moreover, we show that in this case $T$ carries a natural CW complex structure that is defined in terms of the intersections of $T$ with its neighbors in the lattice tiling $\{T+z\,:\, z\in \mathbb{Z}^3\}$ induced by $T$. This CW complex structure is isomorphic to the CW complex defined by the truncated octahedron. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_12076 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On self-affine tiles that are homeomorphic to a ball Thuswaldner, Jörg M. Zhang, Shu-Qin Geometric Topology Primary: 28A80, 57M50. Secondary: 51M20, 52C22, 54F65 Let $M$ be a $3\times 3$ integer matrix which is expanding in the sense that each of its eigenvalues is greater than $1$ in modulus and let $\mathcal{D} \subset \mathbb{Z}^3$ be a digit set containing $|\det M|$ elements. Then the unique nonempty compact set $T=T(M,\mathcal{D})$ defined by the set equation $MT=T+\mathcal{D}$ is called an integral self-affine tile if its interior is nonempty. If $\mathcal{D}$ is of the form $\mathcal{D}=\{0,v,\ldots, (|\det M|-1)v\}$ we say that $T$ has a collinear digit set. The present paper is devoted to the topology of integral self-affine tiles with collinear digit sets. In particular, we prove that a large class of these tiles is homeomorphic to a closed $3$-dimensional ball. Moreover, we show that in this case $T$ carries a natural CW complex structure that is defined in terms of the intersections of $T$ with its neighbors in the lattice tiling $\{T+z\,:\, z\in \mathbb{Z}^3\}$ induced by $T$. This CW complex structure is isomorphic to the CW complex defined by the truncated octahedron. |
| title | On self-affine tiles that are homeomorphic to a ball |
| topic | Geometric Topology Primary: 28A80, 57M50. Secondary: 51M20, 52C22, 54F65 |
| url | https://arxiv.org/abs/2107.12076 |