首页/论文/2011.12880

arXiv 2011.128802020-11math.DSmath.NT

欧几里得空间之外的纯点衍射与熵

Pure point diffraction and entropy beyond the Euclidean space

Till Hauser

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

摘要英文摘要相关论文

AI总结：

本文将纯点衍射 Delone 集的零熵性质推广到 σ-紧局部紧 Abel 群，证明相关动力系统拓扑熵为零，并构造反例说明补片计数熵可能非零、不可沿极限计算甚至无限。

摘要：

对于具有有限局部复杂度和均匀补片频率的欧几里得纯点衍射 Delone 集，众所周知，沿闭中心球计算的补片计数熵为零。我们在 σ-紧局部紧 Abel 群的背景下考虑此类集合，并证明相关 Delone 动力系统的拓扑熵为零。为此，我们给出了一个适当版本的变分原理。我们还构造了反例，表明在此背景下此类集合的补片计数熵可以非零。另一些反例将表明，在此设定下，此类集合的补片计数熵不能沿某个极限计算，甚至可能是无限的。

英文摘要：

For Euclidean pure point diffractive Delone sets of finite local complexity and with uniform patch frequencies it is well known that the patch counting entropy computed along the closed centred balls is zero. We consider such sets in the setting of sigma-compact locally compact Abelian groups and show that the topological entropy of the associated Delone dynamical system is zero. For this we provide a suitable version of the variational principle. We furthermore construct counterexamples, which show that the patch counting entropy of such sets can be non-zero in this context. Other counterexamples will show that the patch counting entropy of such a set can not be computed along a limit and even be infinite in this setting.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

摘要英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra

Certain topological dynamical systems are considered that arise from actions of $σ$-compact locally compact Abelian groups on compact spaces of translation bounded measures. Such a measure dynamical system is shown to have pure point dynamical spectrum if and only if its diffraction spectrum is pure point.

2004-04-07math.DS

Pure Point Dynamical and Diffraction Spectra

We show that for multi-colored Delone point sets with finite local complexity and uniform cluster frequencies the notions of pure point diffraction and pure point dynamical spectrum are equivalent.

2009-10-26math-phmath.DSmath.MP

欧氏空间与双曲空间上随机测度的超均匀性

Hyperuniformity of random measures on Euclidean and hyperbolic spaces

该研究探究欧氏与实双曲空间上不变局部平方可积随机测度的数方差渐近下界，推广了欧氏情形的经典结果，证明双曲空间随机测度无几何超均匀性并定义了谱超均匀性与隐身性。

2024-05-21math.CAmath.DSmath.PR

Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies

Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity function satisfies a certain growth restriction.

2007-09-14math.COmath.DS

Deformation of Delone dynamical systems and pure point diffraction

This paper deals with certain dynamical systems built from point sets and, more generally, measures on locally compact Abelian groups. These systems arise in the study of quasicrystals and aperiodic order, and important subclasses of them exhibit pure point diffraction spectra. We discuss the relevant framework and recall fundamental results and examples. In particular, we show that pure point diffraction is stable under ``equivariant'' local perturbations and discuss various examples,including deformed model sets. A key step in the proof of stability consists in transforming the problem into …

2005-01-18math-phmath.DSmath.MP

Pure point diffraction and cut and project schemes for measures: The smooth case

We present cut and project formalism based on measures and continuous weight functions of sufficiently fast decay. The emerging measures are strongly almost periodic. The corresponding dynamical systems are compact groups and homomorphic images of the underlying torus. In particular, they are strictly ergodic with pure point spectrum and continuous eigenfunctions. Their diffraction can be calculated explicitly. Our results cover and extend corresponding earlier results on dense Dirac combs and continuous weight functions with compact support. They also mark a clear difference in terms of facto…

2006-03-18math.CAmath.DS

更多相关论文收起更多论文

Planar dynamical systems with pure Lebesgue diffraction spectrum

We examine the diffraction properties of lattice dynamical systems of algebraic origin. It is well-known that diverse dynamical properties occur within this class. These include different orders of mixing (or higher-order correlations), the presence or absence of measure rigidity (restrictions on the set of possible shift-invariant ergodic measures to being those of algebraic origin), and different entropy ranks (which may be viewed as the maximal spatial dimension in which the system resembles an i.i.d.\ process). Despite these differences, it is shown that the resulting diffraction spectra a…

2010-06-11math-phmath.DSmath.MP

Random fields on model sets with localized dependency and their diffraction

For a random field on a general discrete set, we introduce a condition that the range of the correlation from each site is within a predefined compact set D. For such a random field omega defined on the model set Lambda that satisfies a natural geometric condition, we develop a method to calculate the diffraction measure of the random field. The method partitions the random field into a finite number of random fields, each being independent and admitting the law of large numbers. The diffraction measure of omega consists almost surely of a pure-point component and an absolutely continuous comp…

2012-09-24math.DS

Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems

There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with a common expansive map $Q$. Our first result gives a partial answer to a question of Lagarias and Wang: we char…

2009-10-23math.DSmath.MG

Pure point/Continuous decomposition of translation-bounded measures and diffraction

In this work we consider translation-bounded measures over a locally compact Abelian group $\mathbb{G}$, with particular interest for their so-called diffraction. Given such a measure $Λ$, its diffraction $\widehatγ$ is another measure on the Pontryagin dual $\widehat{\mathbb{G}}$, whose decomposition into the sum $\widehatγ = \widehatγ_{\mathrm{p}}+ \widehatγ_{\mathrm{c}}$ of its atomic and continuous parts is central in diffraction theory. The problem we address here is whether the above decomposition of $\widehatγ$ lifts to $Λ$ itself, that is to say, whether there exists a decomposition $Λ…

2016-03-29math-phmath.DSmath.MP

Aperiodic order via dynamical systems: Diffraction for sets of finite local complexity

We give an introduction into diffraction theory for aperiodic order. We focus on an approach via dynamical systems and the phenomenon of pure point diffraction. We review recent results and sketch proofs. We then present a new uniform Wiener/Wintner type result generalizing various earlier results of this type.

2007-12-09math-phmath.DSmath.MP

Diffraction of a model set with complex windows

The well-known plastic number substitution gives rise to a ternary inflation tiling of the real line whose inflation factor is the smallest Pisot-Vijayaraghavan number. The corresponding dynamical system has pure point spectrum, and the associated control point sets can be described as regular model sets whose windows in two-dimensional internal space are Rauzy fractals with a complicated structure. Here, we calculate the resulting pure point diffraction measure via a Fourier matrix cocycle, which admits a closed formula for the Fourier transform of the Rauzy fractals, via a rapidly converging…

2019-04-24math-phmath.DSmath.MP

Close-packed dimers on the line: diffraction versus dynamical spectrum

The translation action of $\RR^{d}$ on a translation bounded measure $ω$ leads to an interesting class of dynamical systems, with a rather rich spectral theory. In general, the diffraction spectrum of $ω$, which is the carrier of the diffraction measure, live on a subset of the dynamical spectrum. It is known that, under some mild assumptions, a pure point diffraction spectrum implies a pure point dynamical spectrum (the opposite implication always being true). For other systems, the diffraction spectrum can be a proper subset of the dynamical spectrum, as was pointed out for the Thue-Morse se…

2011-03-08math-phmath.DSmath.MP

Aperiodic order and spherical diffraction, III: The shadow transform and the diffraction formula

We define spherical diffraction measures for a wide class of weighted point sets in commutative spaces, i.e. proper homogeneous spaces associated with Gelfand pairs. In the case of the hyperbolic plane we can interpret the spherical diffraction measure as the Mellin transform of the auto-correlation distribution. We show that uniform regular model sets in commutative spaces have a pure point spherical diffraction measure. The atoms of this measure are located at the spherical automorphic spectrum of the underlying lattice, and the diffraction coefficients can be characterized abstractly in ter…

2020-02-13math.DSmath.GRmath.RT

Pure point diffractive substitution Delone sets have the Meyer property

We prove that a primitive substitution Delone set, which is pure point diffractive, is a Meyer set. This answers a question of J. C. Lagarias. We also show that for primitive substitution Delone sets, being a Meyer set is equivalent to having a relatively dense set of Bragg peaks. The proof is based on tiling dynamical systems and the connection between the diffraction and dynamical spectra.

2006-05-05math.DS

Substitution Delone Sets with Pure Point Spectrum are Inter Model Sets

The paper establishes an equivalence between pure point diffraction and certain types of model sets, called inter model sets, in the context of substitution point sets and substitution tilings. The key ingredients are a new type of coincidence condition in substitution point sets, which we call algebraic coincidence, and the use of a recent characterization of model sets through dynamical systems associated with the point sets or tilings.

2009-10-23math.DSmath.MG

Pure Discrete Spectrum and Regular Model Sets in Unimodular Substitution Tilings on R^d

We consider primitive substitution tilings on R^d whose expansion maps are unimodular. We assume that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, we can construct a cut-and-project scheme with a Euclidean internal space. Under some additional condition, we show that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.

2020-07-22math.DS

On arithmetic progressions in model sets

In this project we show the existence of arbitrary length arithmetic progressions in model sets and Meyer sets in the Euclidean $d$-space. We prove a van der Waerden type theorem for Meyer sets. We show that pure point subsets of Meyer sets with positive density and pure point diffraction contain arithmetic progressions of arbitrary length.

2020-03-30math-phmath.DSmath.MP