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 in the spectrum. 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 and allows explicit calculations also in cases where one meets regular model sets with Rauzy fractals as windows.Dedicated to Manfred Denker on the occasion of his 75th birthday

Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction

“…It turns out that all 48 DPV inflation tilings are regular model sets, and hence are pure point diffractive; see Theorem 5.2 in Baake et al (2021). They all share the same Fourier module, L Ã Â L Ã .…”

“…The matrix B is obtained by first taking the ?-map of the setvalued displacement matrix T of equation 4and then its inverse Fourier transform. For this reason, B is called the internal Fourier matrix (Baake & Grimm, 2019b), to distinguish it from the Fourier matrix of the renormalization approach in physical space (Baake & Gä hler, 2016;Baake, Frank et al, 2019); see Bufetov & Solomyak (2018, 2020 for various extensions with more flexibility in the choice of the interval lengths. In Dirac notation, we set jhi ¼ ðh a ; h b Þ T , which satisfies jhð0Þi ¼ jvi with the right eigenvector jvi of the substitution matrix M from equation 2.…”

“…In turn, this means that the internal cocycle approach applies and can be used to compute the Fourier transforms and the diffraction intensities for such tilings; see Baake & Grimm (2020) for details. Here, we discuss examples of planar projection tilings with fractally bounded windows, which are based on direct product variations (DPVs) (Sadun, 2008;Frank, 2015) of Fibonacci systems, as recently described by Baake et al (2021). Clearly, if one considers a direct product structure based on the Fibonacci tiling, one obtains a tiling of the plane, called the square Fibonacci tiling.…”

“…In particular, 20 of these DPVs possess windows of Rauzy fractal type, of which there are three different types, called 'castle', 'cross' and 'island' by Baake et al (2021). They have different fractal dimension of the window boundaries, which are approximately 1.875, 1.756 and 1.561, respectively.…”

“…While systems based on a cut-and-project scheme are generally well understood, this is less so for systems originating from substitution or inflation rules, which constitute another popular method of generating systems with aperiodic order; see Queffé lec (2010), , Frettlö h (2017), and references therein for details. There has been recent progress particularly on substitutions of constant length; see Mañ ibo (2017), Bartlett (2018), Berlinkov & Solomyak (2019), , Baake, Frank et al (2019), Bufetov & Solomyak (2020).…”