Trace and antitrace maps for aperiodic sequences: Extensions and applications

We study aperiodic systems based on substitution rules by means of a transf er-matrix approach. In addition to the well-known trace map, we investigate the so-called "antitrace" map, which is the corresponding map for the diff erence of the off-diagonal elements of the 2X2 transfer matrix. The antitra ce maps are obtained for various binary, ternary, and quaternary aperiodic sequences, such as the Fibonacci, Thue-Morse, period-doubling, Rudin-Shapir o sequences, and certain generalizations. For arbitrary substitution rules, we show that not only trace maps, but also antitrace maps exist The dimens ion of our antitrace map is r(r+1)/2, where r denotes the number of basic l etters in the aperiodic sequence. Analogous maps for specific matrix elemen ts of the transfer matrix can also be constructed, but the maps for the off -diagonal elements and for the difference of the diagonal elements coincide with the antitrace map. Thus, from the trace and antitrace map, we Can det ermine any physical quantity related to the global transfer matrix of the s ystem. As examples, we employ these dynamical maps to compute the transmiss ion coefficients for optical multilayers, harmonic chains, and electronic s ystems.