# Cut-and-Project Schemes for Pisot Family Substitution Tilings

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary

#### 2.1. Tilings

#### 2.2. Delone $\kappa $-Sets

#### 2.3. Substitutions

**Definition**

**1.**

#### 2.4. Cut-and-Project Scheme

**Definition**

**2.**

#### 2.5. Pure Point Spectrum

## 3. Cut-and-Project Scheme for Pisot Family Substitution Tilings

**Definition**

**3**

**.**Let $\mathcal{T}$ be a fixed point of a primitive substitution with expansion map ϕ. For every $\mathcal{T}$-tile T, we choose a tile $\gamma T$ on the patch $\omega \left(T\right)$. For all tiles of the same type, we choose $\gamma T$ with the same relative position. This defines a map $\gamma :\mathcal{T}\to \mathcal{T}$ called the tile map. Then, we define the control point for a tile $T\in \mathcal{T}$ by

- (a)
- ${T}^{\prime}=T+c\left({T}^{\prime}\right)-c\left(T\right)$, for any tiles $T,{T}^{\prime}$ of the same type;
- (b)
- $\varphi \left(c\right(T\left)\right)=c\left(\gamma T\right)$, for $T\in \mathcal{T}$.

**Theorem**

**1**

**.**Let $\mathcal{T}$ be a primitive substitution tiling on ${\mathbb{R}}^{d}$ with expansion map ϕ. We assume that $\mathcal{T}$ has FLC, ϕ is diagonalizable, and all the eigenvalues of ϕ are algebraically conjugate with multiplicity one. Then, there exists an isomorphism $\sigma :{\mathbb{R}}^{d}\to {\mathbb{R}}^{d}$ such that

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

## 4. Two Cut-and-Project Schemes

#### 4.1. $\varphi $-Topology

**Lemma**

**3**

**.**Let $\mathcal{T}$ be a primitive substitution tiling with an expansive map ϕ. Suppose that $\mathcal{T}$ admits an algebraic coincidence. Then, the system $\{{\varphi}^{n}\Xi \left(\mathcal{T}\right)+\mathcal{K}:n\in {\mathbb{Z}}_{+}\}$ serves as a neighbourhood base for $0\in L$ of the topology on L relative to which L is a topological group.

#### 4.2. ${P}_{\u03f5}$-Topology

**Proposition**

**1**

**.**Let $\mathcal{T}$ be a primitive substitution tiling. Assume an algebraic coincidence on $\mathcal{T}$, then the map $\iota :x\mapsto x$ from ${L}_{\varphi}$ onto ${L}_{P}$ is topologically isomorphic.

**Remark**

**1.**

**Theorem**

**2.**

**Proof.**Since $\varphi $ is an expansive map and satisfies the Pisot family condition, we first note that there is no algebraic conjugate $\gamma $ of eigenvalues of $\varphi $ with $\left|\gamma \right|=1$.

**Theorem**

**3**

**.**Let $\mathcal{T}$ be a primitive substitution tiling in $\mathbb{R}$ with expansion factor β being a unimodular irreducible Pisot number. Then $\mathcal{T}$ has a pure point spectrum if and only if for any $1\le i\le \kappa $, each ${\mathcal{C}}_{i}$ is a regular model set in CPS (10).

**Corollary**

**1.**

**Proof.**By Theorem 3, it is known that for a primitive Pisot substitution tiling in $\mathbb{R}$, if $\mathcal{T}$ has a pure point spectrum, then $\mathcal{C}$ is a regular model $\kappa $-set in CPS (10).

## 5. Conclusions

## 6. Further Study

**Question**

**1.**

**Question**

**2.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Lee, J.-Y.; Akiyama, S.; Nagai, Y.
Cut-and-Project Schemes for Pisot Family Substitution Tilings. *Symmetry* **2018**, *10*, 511.
https://doi.org/10.3390/sym10100511

**AMA Style**

Lee J-Y, Akiyama S, Nagai Y.
Cut-and-Project Schemes for Pisot Family Substitution Tilings. *Symmetry*. 2018; 10(10):511.
https://doi.org/10.3390/sym10100511

**Chicago/Turabian Style**

Lee, Jeong-Yup, Shigeki Akiyama, and Yasushi Nagai.
2018. "Cut-and-Project Schemes for Pisot Family Substitution Tilings" *Symmetry* 10, no. 10: 511.
https://doi.org/10.3390/sym10100511