Rigorous Results for Aperiodic and Almost Periodic Substitution Sequences

Recent studies of the eects of geometric uctuations, associated with aperiodic exchange inter-actions, on the critical behavior of ferromagnetic Ising models on hierarchical lattices, were the motivation to investigate some properties of two-letter sequences generated by uniform substitu-tions. We rigorously identify the substitutions that generate either periodic or aperiodic (almost periodic or not) sequences. For instance, two known substitutions, the Thue-Morse and the period-doubling rules, generate aperiodic sequences of almost periodic type.