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Renormalisation of Pair Correlation Measures for Primitive Inflation Rules and Absence of Absolutely Continuous Diffraction

  • Michael BaakeEmail author
  • Franz Gähler
  • Neil Mañibo
Article

Abstract

The pair correlations of primitive inflation rules are analysed via their exact renormalisation relations. We introduce the inflation displacement algebra that is generated by the Fourier matrix of the inflation and deduce various consequences of its structure. Moreover, we derive a sufficient criterion for the absence of absolutely continuous diffraction components, as well as a necessary criterion for its presence. This is achieved via estimates for the Lyapunov exponents of the Fourier matrix cocycle of the inflation rule. We also discuss some consequences for the spectral measures of such systems. While we develop the theory first for the classic setting in one dimension, we also present its extension to primitive inflation rules in higher dimensions with finitely many prototiles up to translations.

Notes

Acknowledgements

It is a pleasure to thank Frederic Alberti, Alan Bartlett, Scott Balchin, Natalie Frank, Uwe Grimm, Andrew Hubery, Robbie Robinson, Boris Solomyak and Nicolae Strungaru for helpful discussions. We also thank two anonymous reviewers for their thoughtful comments. This work was supported by the German Research Foundation (DFG), within the CRC 1283.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany

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