2023/11/09 by Neil Mañibo, Mañibo, Neil
Materials Science · #37B10 #52C23 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2311.05731
openalex publication_date 2023/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Tilings and point sets arising from substitutions are classical mathematical models of quasicrystals. Their hierarchical structure allows one to obtain concrete answers regarding spectral questions tied to the underlying measures and potentials. In this review, we present some generalisations of substitutions, with a focus on substitutions on compact alphabets, and with an outlook towards their spectral theory. Guided by two main examples, we will illustrate what changes when one moves from finite to compact (infinite) alphabets, and discuss under which assumptions do we recover the usual geometric and statistical properties which make them viable models of materials with almost periodic order. We also present a planar example (which is a two-dimensional generalisation of the Thue--Morse substitution), whose diffraction is purely singular continuous.