Pure point spectrum and regular model sets in substitution tilings on ℝd

2021 ◽  
pp. 699-700
Author(s):  
Jeong-Yup Lee
Keyword(s):  
Point Spectrum ◽  
Pure Point ◽  
Pure Point Spectrum ◽  
Regular Model ◽  
Model Sets ◽  
Substitution Tilings

scholarly journals Cut-and-Project Schemes for Pisot Family Substitution Tilings

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 511 ◽  
Author(s):  
Jeong-Yup Lee ◽  
Shigeki Akiyama ◽  
Yasushi Nagai
Keyword(s):  
Point Spectrum ◽  
The Other ◽  
Pure Point ◽  
Pure Point Spectrum ◽  
Internal Space ◽  
Sufficient Condition ◽  
Abstract Space ◽  
Dynamical Spectrum ◽  
Substitution Tiling ◽  
Substitution Tilings

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.

scholarly journals Cut-and-Project Schemes for Pisot Family Substitution Tilings

2018 ◽  
Author(s):  
Jeong-Yup Lee ◽  
Shigeki Akiyama ◽  
Yasushi Nagai
Keyword(s):  
Point Spectrum ◽  
The Other ◽  
Pure Point ◽  
Pure Point Spectrum ◽  
Internal Space ◽  
Sufficient Condition ◽  
Abstract Space ◽  
Dynamical Spectrum ◽  
Substitution Tiling ◽  
Substitution Tilings

We consider Pisot family substitution tilings in $\R^d$ whose dynamical spectrum is pure point. There are two cut-and-project schemes(CPS) which arise naturally: one from the Pisot family property and the other from the pure point spectrum respectively. The first CPS has an internal space $\R^m$ for some integer $m \in \N$ defined from the Pisot family property, and the second CPS has an internal space $H$ which is an abstract space defined from the property of the pure point spectrum. However it is not known how these two CPS's are related. Here we provide a sufficient condition to make a connection between the two CPS's. In the case of Pisot unimodular substitution tiling in $\R$, the two CPS's turn out to be same due to [5, Remark 18.5].

scholarly journals Three variations on a theme by Fibonacci

2020 ◽  
pp. 2140001
Author(s):  
Michael Baake ◽  
Natalie Priebe Frank ◽  
Uwe Grimm
Keyword(s):  
Direct Product ◽  
Two Dimensions ◽  
Pure Point ◽  
One Dimension ◽  
Stochastic Nature ◽  
Regular Model ◽  
Model Sets ◽  
Planar Extension ◽  
Dynamical Spectra ◽  
Variations On A Theme

Several variants of the classic Fibonacci inflation tiling are considered in an illustrative fashion, in one and in two dimensions, with an eye on changes or robustness of diffraction and dynamical spectra. In one dimension, we consider extension mechanisms of deterministic and of stochastic nature, while we look at direct product variations in a planar extension. For the pure point part, we systematically employ a cocycle approach that is based on the underlying renormalization structure. It allows explicit calculations, particularly in cases where one meets regular model sets with Rauzy fractals as windows.

scholarly journals Pure Point Spectrum of the Laplacians on Fractal Graphs

1995 ◽  
Vol 129 (2) ◽  
pp. 390-405 ◽  
Author(s):  
L. Malozemov ◽  
A. Teplyaev
Keyword(s):  
Point Spectrum ◽  
Pure Point ◽  
Pure Point Spectrum
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