The properties of one-dimensional quasiperiodic lattice's phonon spectrum

A numerical method with renormalization group transformation is used to study the scaling properties of phonon spectrum and its relevant state of one-dimensional quasiperiodic lattice which is constructed by reduced map. We find that the phonon spectrum at finite gaps' edges in the binary chain Fibonacci model is a Cantor-like set spectrum. The spectrum is singularly continuous and the state is a critical state.