  We construct Sierpinski-carpet (SC) based on AA or AB bilayer graphene by
atom vacancies, namely, SC-AA and SC-AB, to investigate the effects of
interlayer coupling on the electronic properties of fractals. Compared with
monolayer graphene SC, their density of states have similar features, such as
Van-Hove singularities and edge states corresponding to the central peaks near
zero energy, but remarkable energy broadening of edge states emerges in
SC-AA(AB). Calculated conductance spectrum shows that the conductance
fluctuations still hold the Hausdorff fractal dimension behavior even with the
interlayer coupling. Thus, the high correlation between quantum conductance and
fractal geometry dimension is not affected by the interlayer coupling in
bilayer graphene SC. We further reveal the quasi-eigenstates in fractal-like
pressure-modulated bilayer graphene, namely, SC-pAA and SC-pAB. Numerical
results show that the density of states of SC-pAA(pAB) show an asymptotic
behavior to those of SC-AA(AB) especially for high energy quasi-eigenstates.
Within a certain energy range, stronger pressure can lead to stronger
localization, forming an efficient fractal space.
