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Overlapping substitutions and tilings

Shigeki Akiyama, Yasushi Nagai, Shu-Qin Zhang

TL;DR

The paper extends substitution tiling theory to overlapping substitutions, addressing non-integer substitution matrices by leveraging weighted substitutions to define tile frequencies via Perron–Frobenius theory. It proves that patch frequencies converge uniformly and are governed by the right PF eigenvector, yielding unique ergodicity for the associated tiling dynamical systems. Under mild assumptions, the expansion constant β is shown to be an algebraic integer, tying geometric inflation to integer PF structure. A consistency criterion based on an open set condition with a graph-directed IFS is developed to ensure illegal overlaps do not arise, with concrete one-dimensional and Delone-set–based higher-dimensional constructions provided. The work lays groundwork for broad construction of overlapping substitutions and poses open problems for further exploration, including connections to Bernoulli convolutions and non-Pisot expansions.

Abstract

We generalize the notion of (geometric) substitution rule to obtain overlapping substitutions. Our motivating example is the substitution presented in Ziherl, Dotera and Bekku \cite{DBZ}, which features a substitution matrix with non-integer entries. We give the meaning of such a matrix by showing that the right Perron--Frobenius eigenvector encodes the patch frequency of the resulting tiling. The patch frequencies are shown to be uniformly convergent, implying that the corresponding dynamical system is uniquely ergodic. Under mild assumptions, we further prove that the associated expansion constant is always an algebraic integer. In general, overlapping substitutions may yield a patch with illegal (partial) overlaps of tiles, even if it is locally consistent. We provide a sufficient condition for an overlapping substitution to be consistent, ensuring that no such illegal tiles emerge. Finally, we construct many intriguing one-dimensional overlapping substitutions and present higher dimensional examples from Delone multi-sets with inflation symmetry.

Overlapping substitutions and tilings

TL;DR

The paper extends substitution tiling theory to overlapping substitutions, addressing non-integer substitution matrices by leveraging weighted substitutions to define tile frequencies via Perron–Frobenius theory. It proves that patch frequencies converge uniformly and are governed by the right PF eigenvector, yielding unique ergodicity for the associated tiling dynamical systems. Under mild assumptions, the expansion constant β is shown to be an algebraic integer, tying geometric inflation to integer PF structure. A consistency criterion based on an open set condition with a graph-directed IFS is developed to ensure illegal overlaps do not arise, with concrete one-dimensional and Delone-set–based higher-dimensional constructions provided. The work lays groundwork for broad construction of overlapping substitutions and poses open problems for further exploration, including connections to Bernoulli convolutions and non-Pisot expansions.

Abstract

We generalize the notion of (geometric) substitution rule to obtain overlapping substitutions. Our motivating example is the substitution presented in Ziherl, Dotera and Bekku \cite{DBZ}, which features a substitution matrix with non-integer entries. We give the meaning of such a matrix by showing that the right Perron--Frobenius eigenvector encodes the patch frequency of the resulting tiling. The patch frequencies are shown to be uniformly convergent, implying that the corresponding dynamical system is uniquely ergodic. Under mild assumptions, we further prove that the associated expansion constant is always an algebraic integer. In general, overlapping substitutions may yield a patch with illegal (partial) overlaps of tiles, even if it is locally consistent. We provide a sufficient condition for an overlapping substitution to be consistent, ensuring that no such illegal tiles emerge. Finally, we construct many intriguing one-dimensional overlapping substitutions and present higher dimensional examples from Delone multi-sets with inflation symmetry.
Paper Structure (14 sections, 16 theorems, 148 equations, 24 figures)

This paper contains 14 sections, 16 theorems, 148 equations, 24 figures.

Table of Contents

  1. Introduction
  2. The theory of weighted substitutions and overlapping substitutions
  3. Definitions of tiles, patches and tilings
  4. The theory of weighted substitutions
  5. The theory of overlapping substitutions
  6. The expanding constant is an algebraic integer
  7. Consistency of pre-overlapping substitution: open set condition
  8. Construction of overlapping substitutions
  9. A construction of one-dimensional overlapping substitutions
  10. Construction in high-dimensions from Delone sets with inflation symmetry
  11. Further problems
  12. Appendix
  13. Notation
  14. Proof

Key Result

Lemma 2.1

For any $T\in\mathcal{B}+\mathbb{R}^d$, there are only finitely many $S\in\mathcal{B}+\mathbb{R}^d$ such that $\xi(S)(T)\neq 0$.

Figures (24)

  • Figure 1: Penrose tiling substitution
  • Figure 2: Ammann-Beenker tiling substitution
  • Figure 3: Square Triangle tiling substitution
  • Figure 4: Bronze-mean tiling substitution
  • Figure 5: DZ substitutions of level $n=1,2,3,4$
  • ...and 19 more figures

Theorems & Definitions (40)

  • Example 1.1: One dimensional symbolic weighted substitution
  • Definition 1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2
  • proof
  • Corollary 2.4
  • Lemma 2.5
  • ...and 30 more