scholarly journals Pure Point Diffractive Substitution Delone Sets Have the Meyer Property

2008 ◽  
pp. 1-20
Author(s):  
Jeong-Yup Lee ◽  
Boris Solomyak
Keyword(s):  
Pure Point ◽  
Delone Sets

scholarly journals Pure Point Diffraction Implies Zero Entropy for Delone Sets with Uniform Cluster Frequencies

2007 ◽  
Vol 82 (1) ◽  
pp. 61-77 ◽  
Author(s):  
Michael Baake ◽  
Daniel Lenz ◽  
Christoph Richard
Keyword(s):  
Pure Point ◽  
Zero Entropy ◽  
Delone Sets

scholarly journals Pure Point Diffractive Substitution Delone Sets Have the Meyer Property

2008 ◽  
Vol 39 (1-3) ◽  
pp. 319-338 ◽  
Author(s):  
Jeong-Yup Lee ◽  
Boris Solomyak
Keyword(s):  
Pure Point ◽  
Delone Sets

scholarly journals A dichotomy for bounded displacement equivalence of Delone sets

2021 ◽  
pp. 1-18
Author(s):  
YOTAM SMILANSKY ◽  
YAAR SOLOMON
Keyword(s):  
Compact Space ◽  
Equivalence Classes ◽  
Natural Topology ◽  
Fell Topology ◽  
Local Complexity ◽  
Substitution Tilings ◽  
Local Topology ◽  
Delone Sets

Abstract We prove that in every compact space of Delone sets in ${\mathbb {R}}^d$ , which is minimal with respect to the action by translations, either all Delone sets are uniformly spread or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty–Fell topology, which is the natural topology on the space of closed subsets of ${\mathbb {R}}^d$ . This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all minimal spaces of Delone sets associated with well-studied constructions such as cut-and-project sets and substitution tilings, whether or not finite local complexity is assumed.

Multidimensional period doubling structures

2016 ◽  
Vol 72 (3) ◽  
pp. 391-394
Author(s):  
Jeong-Yup Lee ◽  
Dvir Flom ◽  
Shelomo I. Ben-Abraham
Keyword(s):  
Arbitrary Dimension ◽  
Period Doubling ◽  
Pure Point ◽  
Doubling Sequence ◽  
Straightforward Extension

This paper develops the formalism necessary to generalize the period doubling sequence to arbitrary dimension by straightforward extension of the substitution and recursion rules. It is shown that the period doubling structures of arbitrary dimension are pure point diffractive. The symmetries of the structures are pointed out.

scholarly journals Dense Dirac combs in Euclidean space with pure point diffraction

2003 ◽  
Vol 44 (10) ◽  
pp. 4436 ◽  
Author(s):  
Christoph Richard
Keyword(s):  
Euclidean Space ◽  
Pure Point

Dynamical Systems with Pure Point Spectrum

1982 ◽  
pp. 325-337
Author(s):  
I. P. Cornfeld ◽  
S. V. Fomin ◽  
Ya. G. Sinai
Keyword(s):  
Dynamical Systems ◽  
Point Spectrum ◽  
Pure Point ◽  
Pure Point Spectrum
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