Three variations on a theme by Fibonacci

TL;DR

This paper explores three variations of Fibonacci inflation tilings in one and two dimensions, analyzing their diffraction and dynamical spectra, and employing a cocycle approach for explicit calculations in regular model set cases.

Contribution

It introduces new variants of Fibonacci tilings and applies a cocycle method to analyze their spectral properties systematically.

Findings

Robustness of diffraction spectra under variations

Explicit calculations for regular model sets with Rauzy fractals

Extension mechanisms influence spectral characteristics

Abstract

Several variants of the classic Fibonacci inflation tiling are considered in an illustrative fashion, in one and in two dimensions, with an eye on changes or robustness of diffraction and dynamical spectra. In one dimension, we consider extension mechanisms of deterministic and of stochastic nature, while we look at direct product variations in a planar extension. For the pure point part, we systematically employ a cocycle approach that is based on the underlying renormalisation structure. It allows explicit calculations, particularly in cases where one meets regular model sets with Rauzy fractals as windows.