[Paper Review] A primer on substitution tilings of the Euclidean plane

Natalie Priebe Frank|arXiv (Cornell University)|May 8, 2007

Quasicrystal Structures and Properties22 references4 citations

TL;DR

This paper introduces a foundational framework for understanding substitution tilings in the Euclidean plane, distinguishing between geometric substitutions (based on linear expansion, as in Penrose tilings) and combinatorial substitutions (based on symbolic concatenation, a newer and less understood class). It establishes connections between these classes, highlights open problems in spectral theory and dynamical systems, and proposes that combinatorial substitutions may give rise to geometric ones under specific conditions, particularly when Pisot eigenvalues are involved.

ABSTRACT

This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include self-similar tilings such as the well-known Penrose tilings; for this class there is a substantial body of research in the literature. Combinatorial substitutions are just beginning to be examined, and some of what we present here is new. We give numerous examples, mention selected major results, discuss connections between the two classes of substitutions, include current research perspectives and questions, and provide an extensive bibliography. Although the author attempts to fairly represent the as a whole, the paper is not an exhaustive survey, and she apologizes for any important omissions.

Motivation & Objective

• To provide a systematic introduction to substitution tilings in the Euclidean plane, categorizing them into geometric and combinatorial classes.
• To clarify the distinction between substitution rules based on linear expansion (geometric) and those based on symbolic concatenation (combinatorial), which lack a unified definition.
• To explore the dynamical and spectral properties of both classes, especially in relation to the Perron eigenvalue and Pisot conditions.
• To identify open problems in the theory, particularly concerning the relationship between combinatorial and geometric substitutions and their dynamical conjugacies.
• To stimulate further research by presenting new examples and posing critical questions on mutual local derivability, spectral theory, and topological conjugacy.

Proposed method

• Classify substitution tilings into two main types: geometric (self-similar, based on linear expansion maps) and combinatorial (symbolic, based on word substitution rules).
• Use symbolic substitution systems (e.g., morphisms on finite alphabets) to model combinatorial substitutions, with examples like the Fibonacci and Chacon substitutions.
• Apply the replace-and-rescale method to generate geometric tilings from combinatorial substitutions, analyzing when the limit yields topologically well-behaved tiles.
• Analyze the substitution matrix and its Perron eigenvalue to assess spectral and dynamical properties, especially in the context of Pisot numbers.
• Examine dual graphs of tilings to compare combinatorial and geometric structures, assessing whether they share topological or combinatorial invariants.
• Investigate topological conjugacies between tiling dynamical systems and their symbolic counterparts, particularly in relation to eigenvalues and suspension dynamics.

Experimental results

Research questions

• RQ1 Under what conditions does a combinatorial substitution give rise to a geometric substitution via the replace-and-rescale method?
• RQ2 To what extent do the spectral properties of combinatorial substitutions mirror those of geometric substitutions, particularly in relation to the Perron eigenvalue and Pisot conditions?
• RQ3 Can topological conjugacy between a combinatorial substitution system and its geometric counterpart be established, and is it compatible with mutual local derivability?
• RQ4 How do the eigenvalues of the substitution matrix influence the dynamical spectrum in both symbolic and tiling systems, especially in the context of continuous suspension actions?
• RQ5 What is the relationship between the dual graphs of combinatorial and geometric tilings, and can combinatorial properties of one inform the structure of the other?

Key findings

• Geometric substitution tilings, such as the Penrose tiling, are well-understood and exhibit self-similarity through linear expansion maps.
• Combinatorial substitutions, though less studied, can generate complex tilings through symbolic substitution rules, with examples like the Fibonacci and Chacon substitutions illustrating non-primitive and primitive cases.
• The presence of a Pisot eigenvalue in the substitution matrix is strongly linked to pure discrete spectrum in geometric substitutions, and this connection is conjectured to extend to combinatorial cases.
• There exist tiling dynamical systems that are topologically conjugate but not mutually locally derivable, indicating that the Curtis-Lyndon-Hedlund theorem does not extend to tiling systems.
• The suspension of a symbolic system into a continuous $×$-action on a tiling space often yields better-understood spectral properties than the discrete action, especially in relation to eigenvalues.
• Dual graphs of combinatorial and geometric tilings share labeled vertices but not necessarily edges or facets, and their structural relationship remains inconsistent and poorly understood.