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We study the structure factor for a large class of sequences of two elements a and b such that longer sequences are generated from shorter ones by a simple substitution rule a → σ 1 ( a, b ) and b → σ 2 ( a, b ), where the σ 's are some sequences of a 's and b 's. Such sequences include periodic and quasiperiodic systems (e.g., the Fibonacci sequence), as well as systems such as the Thue-Morse sequence, which are neither. We show that there are values of the frequency ω at which the structure factors of these sequences have peaks that scale with L , the size of the system like L α(ω) . For a given sequence a simple one- or two-dimensional dynamical iterative map of the variable ω can easily be abstracted from the substitution algorithm. The basin of attraction of a given fixed point or limit cycle of this map is a set of values of ω at which there are peaks of the structure factor all of which share the same value of α . Furthermore, only those values of ω which are in the basin of attraction of the origin can have α ( ω )=2. All other peaks will grow less rapidly with L . We show how to construct many sequences which, like the Thue-Morse sequence, have no L 2 peaks. Other qualitative features of the structure factors are presented. Our approach unifies the treatment of a large class of apparently very diverse systems. Implications for the band structure of these systems as well as for the analysis of sequences with more than two elements are discussed

Structure factor of substitutional sequences