Renormalisation of pair correlation measures for primitive inflation rules and absence of absolutely continuous diffraction

TL;DR

This paper develops a renormalisation framework for analyzing pair correlations in primitive inflation rules, providing criteria for the absence or presence of absolutely continuous diffraction, with implications for spectral measures in multiple dimensions.

Contribution

It introduces the inflation displacement algebra and derives new criteria for diffraction types, extending the theory to higher-dimensional inflation rules.

Findings

Criteria for absence of absolutely continuous diffraction.

Extension of the theory to higher dimensions.

Analysis of spectral measures for inflation systems.

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.