On self-affine tiles whose boundary is a sphere
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in: Transactions of the American Mathematical Society, Jahrgang 373.2020, Nr. 1, 01.2020, S. 491-527.
Publikationen: Beitrag in Fachzeitschrift › Artikel › Forschung › (peer-reviewed)
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TY - JOUR
T1 - On self-affine tiles whose boundary is a sphere
AU - Thuswaldner, Jörg
AU - Zhang, Shuqin
N1 - Publisher Copyright: © 2019 American Mathematical Society
PY - 2020/1
Y1 - 2020/1
N2 - Let M be a 3×3 integer matrix each of whose eigenvalues is greater than 1 in modulus and let D ⊂ Z 3 be a set with |D| = | det M|, called a digit set. The set equation MT = T + D uniquely defines a nonempty compact set T ⊂ 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 D = {0, v, 2v, . . ., (| det M| − 1)v} for some v ∈ Z 3 \ {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.
AB - Let M be a 3×3 integer matrix each of whose eigenvalues is greater than 1 in modulus and let D ⊂ Z 3 be a set with |D| = | det M|, called a digit set. The set equation MT = T + D uniquely defines a nonempty compact set T ⊂ 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 D = {0, v, 2v, . . ., (| det M| − 1)v} for some v ∈ Z 3 \ {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.
KW - Self-affine sets
KW - tiles and tilings
KW - Low dimensional topology
KW - truncated octahedron
UR - http://www.scopus.com/inward/record.url?scp=85077396401&partnerID=8YFLogxK
U2 - 10.1090/tran/7930
DO - 10.1090/tran/7930
M3 - Article
VL - 373.2020
SP - 491
EP - 527
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
SN - 0002-9947
IS - 1
ER -