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On self-affine tiles that are homeomorphic to a ball

Jörg M. Thuswaldner, Shu-Qin Zhang

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.

On self-affine tiles that are homeomorphic to a ball

Abstract

Let be a integer matrix which is expanding in the sense that each of its eigenvalues is greater than in modulus and let be a digit set containing elements. Then the unique nonempty compact set defined by the set equation is called an integral self-affine tile if its interior is nonempty. If is of the form we say that 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 -dimensional ball. Moreover, we show that in this case carries a natural CW complex structure that is defined in terms of the intersections of with its neighbors in the lattice tiling induced by . This CW complex structure is isomorphic to the CW complex defined by the truncated octahedron.

Paper Structure

This paper contains 20 sections, 26 theorems, 72 equations, 9 figures.

Table of Contents

  1. introduction
  2. Context of the paper
  3. Description of the main results
  4. Outline of the paper
  5. Intersections of self-affine tiles and CW complexes
  6. Basic properties of self-affine tiles
  7. A normal form
  8. The neighbor graph
  9. The Hata graph and Peano continua
  10. On the topology of certain subsets of $\partial T$
  11. Types of intersections
  12. Basic properties of intersections
  13. The type of an intersection
  14. A graph that governs the types of intersections
  15. Proofs of the main results
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $T=T(M,\mathcal{D})$ be a $3$-dimensional self-affine tile with collinear digit set and assume that the characteristic polynomial $\chi(x)=x^3+Ax^2+Bx+C$ of $M$ satisfies $1 = A\le B < C$. If $T$ has $14$ neighbors then $T$ is a $3$-ball.

Figures (9)

  • Figure 1: Two examples of $3$-dimensional self-affine tiles.
  • Figure 2: The CW complex structure of a self-affine tile $T$.
  • Figure 3: The neighbor graph $G(\mathcal{S})$ for an $ABC$-tile $T$ with $1\le A \le B<C$ having $14$ neighbors. Here we set $P=(1,0,0)^t, ~Q=(A,1,0)^t,~ N=(B,A,1)^t$. To save space we write $\alpha\xrightarrow{e}\alpha'$ instead of $\alpha\xrightarrow{(e)_M}\alpha'$ in this figure (recall the notation \ref{['eq:Mnotation']}). Multiple labels correspond to multiple edges. If an edge has labels $d,\ldots, d'$ with $d>d'$ then the edge has to be deleted.
  • Figure 4: The Hata graph $H(\mathcal{S})$ (left) which is isomorphic to the graph of vertices and edges of the so-called tetrakis hexahedron. The tetrakis hexahedron (right) is a Catalan polyhedron which is the dual of the truncated octahedron (see e.g.CBG:08).
  • Figure 5: The Hata graph $H_P$ (we omit the multiplication by $M^{-1}$ and write $e$ instead of $(e)_M$ to save space).
  • ...and 4 more figures

Theorems & Definitions (58)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1
  • Definition 2.2: Neighbor graph; cf. ScheicherThuswaldner03
  • Remark 2.3
  • Lemma 2.4
  • ...and 48 more