On weakly almost periodic measures

Lenz, Daniel; Strungaru, Nicolae

doi:10.1090/tran/7422

Cited by 23 publications

(42 citation statements)

References 74 publications

(136 reference statements)

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“…While the explicit decomposition of such measure is in general hard to obtain, by using Perron-Frobenius theory for mixing systems, we give such a decomposition in (2), also see Theorem 3.5. Also, by the results of [20], we have that weighted return time measures for mixing transformations are, in general, not weakly almost periodic. Other works where the autocorrelation of group actions are investigated include [19,24,26], and works discussing the effect of mixing conditions on the autocorrelation of tilings include [7,23,32].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 78%

“…We remark that in [5], Baake and Lenz, using the work of Gouéré [11], constructed a natural autocorrelation on the space of translation bounded measures under group actions. In [20] weakly almost periodic measures are considered, and shown to have a unique decomposition into a pure point diffractive part and continuous diffractive part. While the explicit decomposition of such measure is in general hard to obtain, by using Perron-Frobenius theory for mixing systems, we give such a decomposition in (2), also see Theorem 3.5.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

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Diffraction of Return Time Measures

et al. 2018

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Letting T denote an ergodic transformation of the unit interval and letting f : [0, 1) → R denote an observable, we construct the f -weighted return time measure µ y for a reference point y ∈ [0, 1) as the weighted Dirac comb with support in Z and weights f • T z (y) at z ∈ Z, and if T is non-invertible, then we set the weights equal to zero for all z < 0. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges from the Fourier transform of its autocorrelation and analyse it for the dependence on the underlying transformation. For certain rapidly mixing transformations and observables of bounded variation, we show that the diffraction of µ y consists of a trivial atom and an absolutely continuous part, almost surely with respect to y. This contrasts what occurs in the setting of regular model sets arising from cut and project schemes and deterministic incommensurate structures. As a prominent example of non-mixing transformations, we consider the family of rigid rotations T α : x → x + α mod 1 with rotation number α ∈ R + . In contrast to when T is mixing, we observe that the diffraction of µ y is pure point, almost surely with respect to y. Moreover, if α is irrational and the observable f is Riemann integrable, then the diffraction of µ y is independent of y. Finally, for a converging sequence (α i ) i∈N of rotation numbers, we provide new results concerning the limiting behaviour of the associated diffractions.Here, for z ∈ Z, we let δ z denote the Dirac point mass at z. In this note we answer the following questions.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 78%

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Diffraction of Return Time Measures

et al. 2018

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“…Recently, the notion of Eberlein convolution was extended to two weakly almost periodic measures in [22].…”

Section: Remark 27mentioning

confidence: 99%

“…These results allow us to study the pure point and continuous spectra, respectively, by studying the components γ s and γ 0 , respectively, of the autocorrelation γ of ω, an idea which was used effectively in many places, (such as [2,3,20,36,37,39,40], to name a few). The particular connection between strong almost periodicity and pure point Fourier transform was also exploited in articles such as [3,4,5,6,7,8,15,16,20,21,22,27,32,33,34,38,41].…”

Section: Introductionmentioning

confidence: 99%

On the Fourier Transformability of Strongly Almost Periodic Measures

Strungaru

2019

Can. J. Math.-J. Can. Math.

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In this paper we characterize the Fourier transformability of strongly almost periodic measures in terms of an integrability condition for its Fourier Bohr series. We also provide a necessary and sufficient condition for a strongly almost periodic measure to be a Fourier transform of a measure. We discuss the Fourier transformability of a measure on R d in terms of its Fourier transform as a tempered distribution. We conclude by looking at a large class of such measures coming from the cut and project formalism.

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“…By using this approach, progress has been made towards understanding the pure point and continuous spectra of measures with lattice support [3,4], and with Meyer set support [30,44,45,48,49] and few examples of measures with FLC support [30,47]. More recently, this decomposition has been studied via spectral decomposition of dynamical systems [2].…”

Section: Introductionmentioning

confidence: 99%