Self-similarities in one-dimensional periodic and quasiperiodic systems.

Abstract

We find hyperinflation rules for periodic and quasiperiodic systems in one dimension which consist of two components and are characterized by a single-parameter \ensuremath{\alpha}. Applying hyperinflation rules, we analyze the diffraction pattern and physical properties described by a class of transfer matrices in SL(2,C). We show that the diffraction pattern is self-similar in the wave-vector--\ensuremath{\alpha} space. We also show that the product of transfer matrices has self-similar structure in its asymptotic behavior in the space spanned by \ensuremath{\alpha} and parameters in the matrices, which gives rise to self-similarity in various physical properties such as transmission coefficient, conductivity, heat conductivity, effective impedance, and spectral diffusion. Possible experiments are also discussed.