TRACE MAPS ASSOCIATED WITH GENERAL 2-LETTER SUBSTITUTION RULES

Spectral properties, as determined by trace maps, of the one-dimensional chains (layered structures) constructed according to general two-letter substitution rules are investigated. In all trace maps thus obtained an important role is played by the quantity I = x2 + y2 + z2 - xyz - 4. However, only a very small fraction of all such trace maps are similar to the Fibonacci golden-mean trace map in that I is their invariant. In addition to the known case of the precious-mean lattices (precious means are ratios of the form 1/2 [m + (m2 + 4) 1/2], m being any positive integer; m = 1 gives the golden mean), we have identified two new large classes of substitution rules that give trace maps with invariant I. One of them is a superset of the precious mean lattices. All other cases represent a vast assortment of different trace maps (and thus the potential for various hitherto unexplored spectral properties) with a unifying feature that the set I = O plays the role of an attractor in the trace space. In most (but not all) cases, two chains with identical trace maps (and thus identical spectra) are locally isomorphic. Generally, local isomorphism equivalence classes seem to be subsets of identical spectrum equivalence classes.