Internal and external quasicrystal inflation center and their scaling factors

The properties of quasicrystals of the cut and project type-namely, self-si milarities or so-called inflation properties-are studied. A complete descri ption is given for centers of the scaling symmetry of a quasicrystal, and t he relevant scaling factors are determined for each "inflation center." If the center is a quasicrystal point, it is called an "internal inflation cen ter"; otherwise, it is an "external" one. It turns out that, for any quasic rystal point u, the set of appropriate scaling factors is a u-dependent one -dimensional quasicrystal. There are infinitely many scaling factors common to all internal inflation centers. The description of external inflation c enters, which are plentiful in any quasicrystal, is a slight modification o f a similar description for the internal ones.