Trace and antitrace maps for aperiodic sequences: Extensions and applications

Abstract

We study aperiodic systems based on substitution rules by means of a transfer-matrix approach. In addition to the well-known trace map, we investigate the so-called ``antitrace'' map, which is the corresponding map for the difference of the off-diagonal elements of the $2\ifmmode\times\else\texttimes\fi{}2$ transfer matrix. The antitrace maps are obtained for various binary, ternary, and quaternary aperiodic sequences, such as the Fibonacci, Thue-Morse, period-doubling, Rudin-Shapiro sequences, and certain generalizations. For arbitrary substitution rules, we show that not only trace maps, but also antitrace maps exist. The dimension of our antitrace map is $r(r+1)/2,$ where r denotes the number of basic letters in the aperiodic sequence. Analogous maps for specific matrix elements 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 determine any physical quantity related to the global transfer matrix of the system. As examples, we employ these dynamical maps to compute the transmission coefficients for optical multilayers, harmonic chains, and electronic systems.