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 -