Pure discrete spectrum and regular model sets in d-dimensional unimodular substitution tilings.

Dong-il Lee,Shigeki Akiyama,Jeong-Yup Lee

doi:10.1107/s2053273320009717

Abstract

Primitive substitution tilings on {\bb R}^d whose expansion maps are unimodular are considered. It is assumed that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, a cut-and-project scheme can be constructed with a Euclidean internal space. Under some additional condition, it is shown that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.

Highlights

In the study of aperiodic tilings, it has been an interesting problem to characterize pure discrete spectrum of tiling dynamical systems (Baake & Moody, 2004)
Given a substitution tiling with pure discrete spectrum, it is known that this can be described via a cut-andproject scheme (CPS) (Lee, 2007)
In the case of one-dimensional substitution tilings with pure discrete spectrum, it is shown that a CPS with a Euclidean internal space can be built and the corresponding representative point sets are regular model sets (Barge & Kwapisz, 2006)

Summary

Introduction

In the study of aperiodic tilings, it has been an interesting problem to characterize pure discrete spectrum of tiling dynamical systems (Baake & Moody, 2004). This property is related to understanding the structure of mathematical quasicrystals. In the case of one-dimensional substitution tilings with pure discrete spectrum, it is shown that a CPS with a Euclidean internal space can be built and the corresponding representative point sets are regular model sets (Barge & Kwapisz, 2006).

Tilings

Delone j-sets

Substitutions

Pure point spectrum and algebraic coincidence

Cut-and-project scheme

Cut-and-project scheme on substitution tilings

A1 Á Á Á

Construction of a cut-and-project scheme

Á Á Á O

Further study

Funding information