Spectral analysis of a family of binary inflation rules

TL;DR

This paper investigates the spectral properties of a family of binary inflation rules, revealing that all such systems exhibit singular diffraction, including pure point and singular continuous types, regardless of their inflation factors.

Contribution

It provides a comprehensive spectral analysis of binary inflation rules with various inflation factors, extending understanding beyond the well-known Fibonacci case.

Findings

All studied inflation rules have singular diffraction.

Diffraction types include pure point and singular continuous.

Results apply to rules with both integer and non-Pisot inflation factors.

Abstract

The family of primitive binary substitutions defined by $1 \mapsto 0 \mapsto 0 1^m$ with $m\in\mathbb{N}$ is investigated. The spectral type of the corresponding diffraction measure is analysed for its geometric realisation with prototiles (intervals) of natural length. Apart from the well-known Fibonacci inflation ($m=1$), the inflation rules either have integer inflation factors, but non-constant length, or are of non-Pisot type. We show that all of them have singular diffraction, either of pure point type or essentially singular continuous.