Abstract

The Dirac combs of primitive Pisot–Vijayaraghavan (PV) inflations on the real line or, more generally, in {mathbb {R}}^d are analysed. We construct a mean-orthogonal splitting for such Dirac combs that leads to the classic Eberlein decomposition on the level of the pair correlation measures, and thus to the separation of pure point versus continuous spectral components in the corresponding diffraction measures. This is illustrated with two guiding examples, and an extension to more general systems with randomness is outlined.

Highlights

  • Symbolic Pisot–Vijayaraghavan (PV) substitutions induce a much-studied class of dynamical systems under the action of Z

  • Of particular interest is the self-similar suspension, which turns the symbolic substitution system into a tiling inflation; see [8, Ch. 4] and references therein, as well as [9,20], for general background. This setting is naturally connected with general inflation tilings in Rd, which is our point of view here

  • We report on some progress in this direction, where we start from the Dirac comb of a PV inflation and split it into two parts, one of which leads to the pure point part of the diffraction measure and the other to the continuous part

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Summary

Introduction

Symbolic Pisot–Vijayaraghavan (PV) substitutions induce a much-studied class of dynamical systems under the action of Z. By means of suitable suspensions, they define natural dynamical systems under the continuous translation action of R. Of particular interest is the self-similar suspension, which turns the symbolic substitution system into a tiling inflation; see [8, Ch. 4] and references therein, as well as [9,20], for general background. This setting is naturally connected with general inflation tilings in Rd , which is our point of view here. As the famous Pisot (or PV) substitution conjecture is still unresolved, despite great effort and progress (see [1] for a summary), it seems a good strategy to consider such systems in a wider setting, where one takes mixed spectra more intofocus

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Preliminaries
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A simple PV inflation with mixed spectrum
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Orthogonality for Eberlein convolution
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Eberlein splitting and decomposition for PV inflations
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Further directions
Eberlein splitting for a lattice gas
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Eberlein splitting for a random inflation
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Full Text
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