Trace maps of two-letter substitution rules are investigated with special emphasis on the underlying algebraic structure and on the existence of invariants. We illustrate the results with the generalized Fibonacci chains and show that the well-known Fricke character I(x, y, z)=x2+y2+z2−2xyz−1 is not the only type of invariant that can occur. We discuss several physical applications to electronic spectra including the gap-labeling theorem, to kicked two-level systems, and to the classical 1D Ising model with non-commuting transfer matrices.
It is shown how root lattices and their reciprocals might serve as the right pool for the construction of quasicrystalline structure models. All non-periodic symmetries observed so far are covered in minimal embedding with maximal symmetry.
The well-known two-dimensional octagonal quasilattice is realized by means of dualization and Klotz construction. We discuss the geometric properties and the extended symmetry of the pattern.The concept of geometric defects is introduced, and an elastic energy measure b E is presented that allows a simple sequencing of the forbidden vertices. After a sketchy comparison with Lennard-Jones calculations, some thermodynamic consequences of bE are discussed. It turns out that the specific heat should show a significant increase in comparison with the crystallographic case.
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