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.

scholarly journals On substitution tilings and Delone sets without finite local complexity

2019 ◽  
Vol 39 (6) ◽  
pp. 3149-3177 ◽  
Author(s):  
Jeong-Yup Lee ◽  
◽  
Boris Solomyak ◽  
◽  
Keyword(s):  
Local Complexity ◽  
Substitution Tilings ◽  
Delone Sets

scholarly journals Diffusion on Delone sets

2017 ◽  
Vol 167 (6) ◽  
pp. 1496-1510
Author(s):  
Sebastian Haeseler ◽  
Xueping Huang ◽  
Daniel Lenz ◽  
Felix Pogorzelski
Keyword(s):  
Delone Sets

scholarly journals Spiral Delone sets and three distance theorem

Nonlinearity ◽  
2020 ◽  
Vol 33 (5) ◽  
pp. 2533-2540 ◽  
Author(s):  
Shigeki Akiyama
Keyword(s):  
Delone Sets

scholarly journals Linearly repetitive Delone sets are rectifiable

2013 ◽  
Vol 30 (2) ◽  
pp. 275-290 ◽  
Author(s):  
José Aliste-Prieto ◽  
Daniel Coronel ◽  
Jean-Marc Gambaudo
Keyword(s):  
Delone Sets

scholarly journals Self Affine Delone Sets and Deviation Phenomena

2017 ◽  
Vol 357 (3) ◽  
pp. 1071-1112 ◽  
Author(s):  
Scott Schmieding ◽  
Rodrigo Treviño
Keyword(s):  
Delone Sets

scholarly journals ON SELF-SIMILARITIES OF CUT-AND-PROJECT SETS

2017 ◽  
Vol 57 (6) ◽  
pp. 430 ◽  
Author(s):  
Zuzana Masáková ◽  
Jan Mazáč
Keyword(s):  
Mathematical Models ◽  
Associative Algebra ◽  
Linear Mapping ◽  
Self Similarity ◽  
Similarity Transformations ◽  
Project Scheme ◽  
Delone Sets

Among the commonly used mathematical models of quasicrystals are Delone sets constructed using a cut-and-project scheme, the so-called cut-and-project sets. A cut-and-project scheme (<em>L</em>,π<sub>1</sub>, π<sub>2</sub>) is given by a lattice <em>L</em> in R<sup>s</sup> and projections π<sub>1</sub>, π<sub>2</sub> to suitable subspaces V<sub>1</sub>, V<sub>2</sub>. In this paper we derive several statements describing the connection between self-similarity transformations of the lattice <em>L</em> and transformations of its projections π<sub>1</sub>(<em>L</em>), π<sub>2</sub>(<em>L</em>). For a self-similarity of a set Σ we take any linear mapping A such that AΣ ⊂ Σ, which generalizes the notion of self-similarity usually restricted to scaled rotations. We describe a method of construction of cut-and-project scheme such that π<sub>1</sub>(<em>L</em>) ⊂ R<sup>2</sup> is invariant under an isometry of order 5. We describe all linear self-similarities of the scheme thus constructed and show that they form an 8-dimensional associative algebra over the ring Z. We perform an example of a cut-and-project set with linear self-similarity which is not a scaled rotation.

Point Sets and Dynamical Systems In the Autocorrelation Topology

2004 ◽  
Vol 47 (1) ◽  
pp. 82-99 ◽  
Author(s):  
Robert V. Moody ◽  
Nicolae Strungaru
Keyword(s):  
Dynamical System ◽  
Point Sets ◽  
Finite Point ◽  
Locally Finite ◽  
Uniform Topology ◽  
Regular Model ◽  
Model Sets ◽  
Project Scheme ◽  
Compact Abelian Groups ◽  
Delone Sets

AbstractThis paper is about the topologies arising from statistical coincidence on locally finite point sets in locally compact Abelian groupsG. The first part defines a uniform topology (autocorrelation topology) and proves that, in effect, the set of all locally finite subsets ofGis complete in this topology. Notions of statistical relative denseness, statistical uniform discreteness, and statistical Delone sets are introduced.The second part looks at the consequences of mixing the original and autocorrelation topologies, which together produce a new Abelian group, the autocorrelation group. In particular the relation between its compactness (which leads then to aG-dynamical system) and pure point diffractivity is considered. Finally for generic regular model sets it is shown that the autocorrelation group can be identified with the associated compact group of the cut and project scheme that defines it. For such a set the autocorrelation group, as aG-dynamical system, is a factor of the dynamical local hull.

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