Three variations on a theme by Fibonacci

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

Spectral theory of spin substitutions

We introduce substitutions in Zm which have non-rectangular domains based on an endomorphism Q of Zm and a set D of coset representatives of Zm/QZm, which we call digit substitutions. Using a finite abelian ‘spin’ group we define ‘spin digit substitutions’ and their subshifts (Σ, Zm). Conditions under which the subshift is measure-theoretically isomorphic to a group extension of an m-dimensional odometer are given, inducing a complete decomposition of the function space L2 (Σ, µ). This enables the use of group characters in Ĝ to derive substitutive factors and analyze the spectra of specific subspaces. We provide general sufficient criteria for the existence of pure point, absolutely continuous, and singular continuous spectral measures, together with some bounds on their spectral multiplicity