The butterflies' effects

This work studies spectral properties of Schr\"odinger operators in the context of aperiodic order, using weighted Delone sets to explore the interplay between the underlying dynamics and spectral properties. We study parameter-dependent families interpolating between periodic and aperiodic regimes, whose spectra form so-called spectral butterflies. These reflect fractal and self-similar structures of the spectra. We review existing results, introduce additional examples, and establish new connections between works in the literature. The framework is largely dimension-independent and extends to non-Abelian groups and more general settings.