Inflation centres of the cut and project quasicrystals

Abstract

A rather general form of the conventional cut and project scheme is used to define quasicrystals as point sets in real n-dimensional Euclidean space. The inflation or, equivalently, the self-similarity properties of such quasicrystals are studied here assuming only the convexity of the acceptance window. Our result is a description of inflation centres of all types in a quasicrystal and a proof that our description is complete: there are no other inflation centres. For any chosen quasicrystal point (`internal inflation centre') u, its inflation properties are given as a set of scaling factors. It turns out that the scaling factors form a one-dimensional quasicrystal with a u-dependent acceptance window (`scaling window'). The intersection of the scaling windows associated with all points of a quasicrystal is the one-dimensional quasicrystal of universal (`internal') scaling symmetries. Its acceptance window is the interval . External inflation centres of a cut and project quasicrystal are those which are not among quasicrystal points. Their complete description is given analogically to the description of the internal ones imposing some additional requirements on the scaling factors. Between any two adjacent quasicrystal points one finds a countable infinity of external inflation centres. The scaling factors belonging to any such centre u form an infinite u-dependent subset of points of the quasicrystal with acceptance window containing .