Three variations on a theme by Fibonacci

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.

arXiv comment22 pages, some typos corrected

References in corpus (4)

• Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction
• Diffraction of stochastic point sets: Explicitly computable examples
• Self-Similar Measures for Quasicrystals
• Renormalisation of pair correlation measures for primitive inflation rules and absence of absolutely continuous diffraction

Cited by in corpus (6)

• Fourier transform of Rauzy fractals and point spectrum of 1D Pisot inflation tilings
• Fibonacci direct product variation tilings
• Inflation versus projection sets in aperiodic systems: The role of the window in averaging and diffraction
• Orbit separation dimension as complexity measure for primitive inflation tilings
• Eberlein decomposition for PV inflation systems
• Pythagorean Triples in the Fibonacci Model Set