A class of self-affine tiles in $\mathbb{R}^d$ that are $d$-dimensional tame balls
| Autor: | Deng, Guotai, Liu, Chuntai, Ngai, Sze-man |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Advances in Mathematics Volume 410, Part A, 3 December 2022, 108716 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.aim.2022.108716 |
| Popis: | We study a family of self-affine tiles in $\mathbb{R}^d$ ($d\ge2$) with noncollinear digit sets, which naturally generalizes a class studied originally by Deng and Lau in $\mathbb{R}^2$ and its extension to $\mathbb{R}^3}$ by the authors. By using Brouwer's invariance of domain theorem, along with a tool which we call horizontal distance, we obtain necessary and sufficient conditions for the tiles to be $d$-dimensional tame balls. This answers positively the conjecture in an earlier paper by the authors stating that a member in a certain class of self-affine tiles is homeomorphic to a $d$-dimensional ball if and only if its interior is connected. Comment: 56 pages, 17 figures |
| Databáze: | arXiv |
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