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 Continuous sums of measures and continuous spectra

1965 ◽  
Vol 7 (1) ◽  
pp. 9-14
Author(s):  
S. Sankaran
Keyword(s):  
Continuous Spectrum ◽  
Hilbert Spaces ◽  
Point Spectrum ◽  
Measurable Space ◽  
The Other ◽  
Pure Point ◽  
Pure Point Spectrum ◽  
Definition Of

Von Neumann's definition of the continuous sum of Hilbert spaces led Segal [3] to define the continuous sum of measures on a measurable space. In this note we employ Segal's definition to investigate the measure structures associated with a self-adjoint transformation of pure point spectrum and a self-adjoint transformation of pure continuous spectrum. While these transformations, as operators on separable Hilbert spaces, are the antithesis of each other we show that in their measure structure one is a particular case of the other.

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
Sign in / Sign up

Export Citation Format

Share Document