Spectral analysis of a family of binary inflation rules

Abstract

The family of primitive binary substitutions defined by $1\mapsto 0\mapsto 01^m$ 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.

20 pages, 1 figure, 1 table; revised version with minor improvements and updates

References in corpus (2)

• Pure Point Dynamical and Diffraction Spectra
• Lyapunov exponents for binary substitutions of constant length

Cited by in corpus (4)

• Scaling of diffraction intensities near the origin: Some rigorous results
• Inflation versus projection sets in aperiodic systems: The role of the window in averaging and diffraction
• Uniformly distributed orbits in ${\mathbb{T}}^d$ and singular substitution dynamical systems
• Absence of absolutely continuous diffraction spectrum for certain S-adic tilings