Asymmetries of Cut-and-Project Sets and Related Tilings

Abstract

In order to characterize the (a)symmetries of cut-and-project sets, we prove the following: any cut-and-project set with the two projections being injective on the lattice is fixed by an affine transformation if and only if (1) the window restricted on the projection of the lattice is fixed by another affine transformation, and (2) both affine transformations induce via the two projections the same transformation on the lattice. By this theorem, we prove that any Pisot tilings are asymmetric with respect to any affine transformations.