scholarly journals Dynamics on the graph of the torus parametrization

2016 ◽  
Vol 38 (3) ◽  
pp. 1048-1085 ◽  
Author(s):  
GERHARD KELLER ◽  
CHRISTOPH RICHARD
Keyword(s):  
Uniform Distribution ◽  
Haar Measure ◽  
Pure Point ◽  
Dynamical Spectrum ◽  
Maximal Density ◽  
Weak Model ◽  
Model Sets ◽  
Push Forward ◽  
Model Set ◽  
Dynamical Properties

Model sets are projections of certain lattice subsets. It was realized by Moody [Uniform distribution in model sets. Canad. Math. Bull. 45(1) (2002), 123–130] that dynamical properties of such a set are induced from the torus associated with the lattice. We follow and extend this approach by studying dynamics on the graph of the map that associates lattice subsets to points of the torus and then we transfer the results to their projections. This not only leads to transparent proofs of known results on model sets, but we also obtain new results on so-called weak model sets. In particular, we prove pure point dynamical spectrum for the hull of a weak model set of maximal density together with the push forward of the torus Haar measure under the torus parametrization map, and we derive a formula for its pattern frequencies.

Uniform Distribution in Model Sets

2002 ◽  
Vol 45 (1) ◽  
pp. 123-130 ◽  
Author(s):  
Robert V. Moody
Keyword(s):  
Uniform Distribution ◽  
Physical Space ◽  
Pure Point ◽  
Internal Space ◽  
Compact Closure ◽  
Related Model ◽  
Model Sets ◽  
Distribution Property ◽  
Distribution Of Points ◽  
Model Set

AbstractWe give a new measure-theoretical proof of the uniform distribution property of points in model sets (cut and project sets). Each model set comes as a member of a family of related model sets, obtained by joint translation in its ambient (the ‘physical’) space and its internal space. We prove, assuming only that the window defining themodel set ismeasurable with compact closure, that almost surely the distribution of points in any model set from such a family is uniform in the sense of Weyl, and almost surely the model set is pure point diffractive.

scholarly journals Design of a structure model set for inorganic compounds based on ping-pong balls linked with snap buttons

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ryo Horikoshi ◽  
Hiroyuki Higashino ◽  
Yoji Kobayashi ◽  
Hiroshi Kageyama
Keyword(s):  
High School Students ◽  
Coordination Compounds ◽  
Periodic Structures ◽  
Inorganic Compounds ◽  
Structure Model ◽  
School Students ◽  
Model Sets ◽  
Ping Pong ◽  
Hands On ◽  
Model Set

Abstract Structure model sets for inorganic compounds are generally expensive; their distribution to all students in a class is therefore usually impractical. We have therefore developed a structure model set to illustrate inorganic compounds. The set is constructed with inexpensive materials: ping-pong balls, and snap buttons. The structure model set can be used to illustrate isomerism in coordination compounds and periodic structures of ceramic perovskites. A hands-on activity using the structure model set was developed for high school students and was well-received by them. Despite the concepts being slightly advanced for them, the students’ retention of the knowledge gained through the activity was tested a week after they completed the activity and was found to be relatively high, demonstrating the usefulness of the activity based on the structure model set.

scholarly journals Periods and factors of weak model sets

2018 ◽  
Vol 229 (1) ◽  
pp. 85-132 ◽  
Author(s):  
Gerhard Keller ◽  
Christoph Richard
Keyword(s):  
Weak Model ◽  
Model Sets

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 Equidistribution of singular measures on nilmanifolds and skew products

2011 ◽  
Vol 31 (6) ◽  
pp. 1785-1817
Author(s):  
FABRIZIO POLO
Keyword(s):  
Haar Measure ◽  
Uniform Density ◽  
Unipotent Group ◽  
Affine Transformations ◽  
Skew Products ◽  
Singular Measures ◽  
Ergodic Properties ◽  
Push Forward ◽  
Multiplicative Ergodic Theorem ◽  
Uniquely Ergodic

AbstractWe prove that for a minimal rotationTon a two-step nilmanifold and any measureμ, the push-forwardTn⋆μofμunderTntends toward Haar measure if and only ifμprojects to Haar measure on the maximal torus factor. For an arbitrary nilmanifold we get the same result along a sequence of uniform density one. These results strengthen Parry’s result [Ergodic properties of affine transformations and flows on nilmanifolds.Amer. J. Math.91(1968), 757–771] that such systems are uniquely ergodic. Extending the work of Furstenberg [Strict ergodicity and transformations of the torus.Amer. J. Math.83(1961), 573–601], we get the same result for a large class of iterated skew products. Additionally we prove a multiplicative ergodic theorem for functions taking values in the upper unipotent group. Finally we characterize limits ofTn⋆μfor some skew product transformations with expansive fibers. All results are presented in terms of twisting and weak twisting, properties that strengthen unique ergodicity in a way analogous to that in which mixing and weak mixing strengthen ergodicity for measure-preserving systems.

scholarly journals Kolakoski-(3, 1) Is a (Deformed) Model Set

2004 ◽  
Vol 47 (2) ◽  
pp. 168-190 ◽  
Author(s):  
Michael Baake ◽  
Bernd Sing
Keyword(s):  
Fixed Point ◽  
Pure Point ◽  
Generic Model ◽  
Diffraction Spectrum ◽  
Substitution Rule ◽  
Primitive Substitution ◽  
Diffraction Property ◽  
Model Set

AbstractUnlike the (classical) Kolakoski sequence on the alphabet {1, 2}, its analogue on {1, 3} can be related to a primitive substitution rule. Using this connection, we prove that the corresponding biin finite fixed point is a regular generic model set and thus has a pure point diffraction spectrum. The Kolakoski-(3, 1) sequence is then obtained as a deformation, without losing the pure point diffraction property.

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 AN APERIODIC TILE WITH EDGE-TO-EDGE ORIENTATIONAL MATCHING RULES

2021 ◽  
pp. 1-29
Author(s):  
James J. Walton ◽  
Michael F. Whittaker
Keyword(s):  
Central Idea ◽  
Dynamical Spectrum ◽  
Matching Rule ◽  
Simple Rules ◽  
New Type ◽  
Aperiodic Tile ◽  
Model Set ◽  
Matching Rules ◽  
Almost All ◽  
Uniquely Ergodic

Abstract We present a single, connected tile which can tile the plane but only nonperiodically. The tile is hexagonal with edge markings, which impose simple rules as to how adjacent tiles are allowed to meet across edges. The first of these rules is a standard matching rule, that certain decorations match across edges. The second condition is a new type of matching rule, which allows tiles to meet only when certain decorations in a particular orientation are given the opposite charge. This forces the tiles to form a hierarchy of triangles, following a central idea of the Socolar–Taylor tilings. However, the new edge-to-edge orientational matching rule forces this structure in a very different way, which allows for a surprisingly simple proof of aperiodicity. We show that the hull of all tilings satisfying our rules is uniquely ergodic and that almost all tilings in the hull belong to a minimal core of tilings generated by substitution. Identifying tilings which are charge-flips of each other, these tilings are shown to have pure point dynamical spectrum and a regular model set structure.

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