On self-affine tiles whose boundary is a sphere

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 $|\mathcal {D}|=|\det M|$, 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 $\mathcal {D}=\{0,v,2v,\ldots , (|\det M|-1)v\}$ 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.