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Transactions of the American Mathematical Society

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On self-affine tiles whose boundary is a sphere


Authors: Jörg Thuswaldner and Shu-qin Zhang
Journal: Trans. Amer. Math. Soc. 373 (2020), 491-527
MSC (2010): Primary 28A80, 57M50, 57N05; Secondary 51M20, 52C22, 54F65
DOI: https://doi.org/10.1090/tran/7930
Published electronically: September 23, 2019
MathSciNet review: 4042883
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Abstract: Let $ M$ be a $ 3\times 3$ integer matrix each of whose eigenvalues is greater than $ 1$ in modulus and let $ \mathcal {D}\subset \mathbb{Z}^3$ be a set with $ \vert\mathcal {D}\vert=\vert\det M\vert$, called a digit set. The set equation $ MT = T+\mathcal {D}$ uniquely defines a nonempty compact set $ T\subset \mathbb{R}^3$. If $ T$ has positive Lebesgue measure it is called a $ 3$-dimensional self-affine tile. In the present paper we study topological properties of $ 3$-dimensional self-affine tiles with collinear digit set, i.e., with a digit set of the form $ \mathcal {D}=\{0,v,2v,\ldots , (\vert\det M\vert-1)v\}$ for some $ v\in \mathbb{Z}^3\setminus \{0\}$. We prove that the boundary of such a tile $ T$ is homeomorphic to a $ 2$-sphere whenever its set of neighbors in a lattice tiling which is induced by $ T$ in a natural way contains $ 14$ elements. The combinatorics of this lattice tiling is then the same as the one of the bitruncated cubic honeycomb, a body-centered cubic lattice tiling by truncated octahedra. We give a characterization of $ 3$-dimensional self-affine tiles with collinear digit set having $ 14$ neighbors in terms of the coefficients of the characteristic polynomial of $ M$. In our proofs we use results of R. H. Bing on the topological characterization of spheres.


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Additional Information

Jörg Thuswaldner
Affiliation: Department of Mathematics and Statistics, University of Leoben, Franz-Josef-Strasse 18, A-8700 Leoben, Austria
Email: joerg.thuswaldner@unileoben.ac.at

Shu-qin Zhang
Affiliation: Department of Mathematics and Statistics, University of Leoben, Franz-Josef-Strasse 18, A-8700 Leoben, Austria
Email: shuqin.zhang@unileoben.ac.at

DOI: https://doi.org/10.1090/tran/7930
Keywords: Self-affine sets, tiles and tilings, low dimensional topology, truncated octahedron
Received by editor(s): November 15, 2018
Received by editor(s) in revised form: June 6, 2019, and June 18, 2019
Published electronically: September 23, 2019
Additional Notes: The authors were supported by FWF project P29910, by FWF-RSF project I3466, and by the FWF doctoral program W1230
Shu-Qin Zhang is the corresponding author
Dedicated: Dedicated to Valérie Berthé on the occasion of her 50$^{th}$ birthday
Article copyright: © Copyright 2019 American Mathematical Society