On self-affine tiles whose boundary is a sphere

Abstract

Let M M be a 3 × 3 3\times 3 integer matrix each of whose eigenvalues is greater than 1 1 in modulus and let D ⊂ Z 3 \mathcal {D}\subset \mathbb {Z}^3 be a set with | D | = | det M | |\mathcal {D}|=|\det M| , called a digit set. The set equation M T = T + D MT = T+\mathcal {D} uniquely defines a nonempty compact set T ⊂ R 3 T\subset \mathbb {R}^3 . If T T has positive Lebesgue measure it is called a 3 3 -dimensional self-affine tile. In the present paper we study topological properties of 3 3 -dimensional self-affine tiles with collinear digit set, i.e., with a digit set of the form D = { 0 , v , 2 v , … , ( | det M | − 1 ) v } \mathcal {D}=\{0,v,2v,\ldots , (|\det M|-1)v\} for some v ∈ Z 3 ∖ { 0 } v\in \mathbb {Z}^3\setminus \{0\} . We prove that the boundary of such a tile T T is homeomorphic to a 2 2 -sphere whenever its set of neighbors in a lattice tiling which is induced by T T in a natural way contains 14 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 3 -dimensional self-affine tiles with collinear digit set having 14 14 neighbors in terms of the coefficients of the characteristic polynomial of M M . In our proofs we use results of R. H. Bing on the topological characterization of spheres.