On self-affine tiles that are homeomorphic to a ball

TL;DR

This paper investigates the topology of certain self-affine tiles in three dimensions, proving many are homeomorphic to a 3-ball and possess a CW complex structure similar to a truncated octahedron.

Contribution

It establishes that a broad class of self-affine tiles with collinear digit sets are topologically 3-balls and describes their CW complex structure.

Findings

Many self-affine tiles are homeomorphic to a 3-dimensional ball.

The CW complex structure of these tiles matches that of a truncated octahedron.

The results apply to tiles generated by expanding integer matrices with collinear digit sets.

Abstract

Let $M$ be a $3\times 3$ integer matrix which is expanding in the sense that each of its eigenvalues is greater than $1$ in modulus and let $\mathcal{D} \subset \mathbb{Z}^3$ be a digit set containing $|\det M|$ elements. Then the unique nonempty compact set $T=T(M,\mathcal{D})$ defined by the set equation $MT=T+\mathcal{D}$ is called an integral self-affine tile if its interior is nonempty. If $\mathcal{D}$ is of the form $\mathcal{D}=\{0,v,\ldots, (|\det M|-1)v\}$ we say that $T$ has a collinear digit set. The present paper is devoted to the topology of integral self-affine tiles with collinear digit sets. In particular, we prove that a large class of these tiles is homeomorphic to a closed $3$-dimensional ball. Moreover, we show that in this case $T$ carries a natural CW complex structure that is defined in terms of the intersections of $T$ with its neighbors in the lattice tiling $\{T+z\,:\, z\in \mathbb{Z}^3\}$ induced by $T$. This CW complex structure is isomorphic to the CW complex defined by the truncated octahedron.