  The Aubry-Andre model is a one-dimensional lattice model for quasicrystals
with localized and delocalized phases. At the localization transition point,
the system displays fractal spectrum, which relates to the Hofstadter
butterfly. In this work, we uncover the exact self-similarity structures in the
energy spectrum. We separate the fractal structures into two parts: the fractal
filling positions of gaps and the scaling of gap sizes. We show that the
fractal fillings emerge for a certain type of incommensurate periodicity
regardless of potential strength. However, the power-law scaling of gap sizes
is characteristic for general incommensurability at the critical point of the
model.
