Fractal structure of spectra in one-dimensional incommensurate systems

The fractal structure of energy spectra is one important property due to self-similarity in incommensurate systems. A fractal splitting rule was reported by Xu (1986) for the one-dimensional Aubry model. The extension of this rule to other more complicated models for one-dimensional incommensurate systems, and even to the on-site Fibonacci quasi-lattice, is studied. The results show that the fractal splitting rule found in the Aubry model may have a much wider range of applicability.