MULTIFRACTAL PROPERTIES OF THE STRUCTURE FACTOR OF A CLASS OF SUBSTITUTIONAL SEQUENCES

We show how to estimate the multifractal generating functional of the structure factor of a class of substitutional sequences. These sequences are sequences of two elements, a and b, and are generated by the repetitive application of the rules a --> sigma-1(a,b) and b --> sigma-2(a,b), with the sigma's consisting of strings of a's and b's. We restrict ourselves to the case in which sigma-1 and sigma-2 each contain R elements. This set includes the case of the Thue-Morse sequence. Subject to a technical assumption, we present a systematic approximation scheme for the multifractal generating functional, in which the lowest-order approximation is the generating functional of an R-scale Cantor set. As a by-product of our analysis, we demonstrate the existence of a discontinuity in the multifractal spectrum of the Thue-Morse sequence, and explain the origin of that discontinuity. We present explicit results for two examples, including the Thue-Morse case, and compare the results of our approximations for the Thue-Morse system with numerical simulations.