Frequency spectrum of a general Vicsek fractal

The nature of the frequency spectrum of a Vicsek fractal of any stage has been predicted using an analysis based on local symmetry and self-similarity. The analysis was carried out for transverse vibrations of the fractal with nearest-neighbor interactions. The complete frequency spectrum was found to contain both nondegenerate and degenerate components. The degenerate component of the frequency spectrum of the nth-stage fractal consists of threefold degenerate, D₂-fold degenerate,..., and Dₙ-fold degenerate modes with D₂=8 and Dₙ=5D_n-1-8 for n{≥}3. In particular, D₂-fold degenerate, D₃-fold degenerate modes,..., etc., are all persistent modes. The term persistent degenerate mode is used to describe a descendant of a parent threefold degenerate mode that repeats itself from one stage to the next at the same frequency. The existence of these persistent modes has been shown to be the consequence of the interplay between the local symmetry and the self-similar nature of the fractal. It has also been shown that these persistent modes are superlocalized in nature.