Pair correlations of aperiodic inflation rules via renormalisation: Some interesting examples

TL;DR

This paper explores the spectral properties of aperiodic inflation rules using renormalisation, analyzing examples like Fibonacci, Thue–Morse, Rudin–Shapiro, and a twisted silver mean chain to understand their pair correlations.

Contribution

It extends the spectral theory of constant length substitutions by deriving renormalisation relations for pair correlations in aperiodic inflation systems and analyzing their spectral consequences.

Findings

Derived renormalisation relations for pair correlations in key examples.

Identified spectral types, including singular spectra, in analyzed systems.

Revisited classical sequences with new geometric and spectral insights.

Abstract

This article presents, in an illustrative fashion, a first step towards an extension of the spectral theory of constant length substitutions. Our starting point is the general observation that the symbolic picture (as defined by the substitution rule) and its geometric counterpart with natural prototile sizes (as defined by the induced inflation rule) may differ considerably. On the geometric side, an aperiodic inflation system possesses a set of exact renormalisation relations for its pair correlation coefficients. Here, we derive these relations for some paradigmatic examples and infer various spectral consequences. In particular, we consider the Fibonacci chain, revisit the Thue--Morse and the Rudin--Shapiro sytem, and finally analyse a twisted extension of the silver mean chain with mixed singular spectrum.