Cut-and-Project Schemes for Pisot Family Substitution Tilings

Jeong-Yup Lee,Yasushi Nagai,Shigeki Akiyama

doi:10.3390/sym10100511

Jeong-Yup Lee, Yasushi Nagai + Show 1 more

Open Access

https://doi.org/10.3390/sym10100511

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Abstract

We consider Pisot family substitution tilings in R d whose dynamical spectrum is pure point. There are two cut-and-project schemes (CPSs) which arise naturally: one from the Pisot family property and the other from the pure point spectrum. The first CPS has an internal space R m for some integer m ∈ N defined from the Pisot family property, and the second CPS has an internal space H that is an abstract space defined from the condition of the pure point spectrum. However, it is not known how these two CPSs are related. Here we provide a sufficient condition to make a connection between the two CPSs. For Pisot unimodular substitution tiling in R , the two CPSs turn out to be same due to the remark by Barge-Kwapisz.

Highlights

The study of long-range aperiodic order has played an important role in understanding the structures of physical models like “quasicrystals”
If C is a regular model κ-set in cut-and-project schemes (CPSs) (10), the internal space H which is the completion of Lφ with φ-topology is isomorphic to the internal space K, which is constructed from using the conjugation map Ψ in (8)
We constructed a natural cut-and-project scheme (10) where the control point sets are in the form of a module Z[φ]ξ

Summary

Introduction

The study of long-range aperiodic order has played an important role in understanding the structures of physical models like “quasicrystals”. As an important class of models for the long-range aperiodic order, substitution tilings have been the subject of a great deal of study Among these substitution tilings, Pisot or Pisot family substitutions get more attention due to their well-ordered properties. There has been a great deal of of study on Pisot substitution sequences or Pisot family substitution tilings on Rd which characterizes the property of pure point spectrum (see [1] and therein). The other CPS is made by constructing an abstract internal space from the property of pure point spectrum [6]. These two CPSs were developed independently from different aims of study.

Tilings

Delone κ-Sets

Substitutions

Cut-and-Project Scheme

Pure Point Spectrum

Cut-and-Project Scheme for Pisot Family Substitution Tilings

Two Cut-and-Project Schemes

Pe -Topology

Conclusions

Further Study