We find a systematic procedure constructing the 2D and 3D " periodic " Penrose tilings, which tend to the infinite Penrose tilings. The electronic states in the 2D Penrose tiling are studied by using this sequence. We observe that the spectral measure is singular continuous in the limit of the infinite size. Most of eigenstates are critical, i.e. neither extended nor localized.
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Tsunetsugu et al. (1986) studied this question.
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