DISPLACIVE TRANSFORMATIONS AND QUASI-CRYSTALLINE SYMMETRIES

The property, for quasiperiodic structures built by the cut and project or any equivalent method, to be related to a periodic lattice by a bounded deformation is a strong restriction on the window defining the strip, or on the atomic surface. We give a sufficient condition (which is also necessary in one dimension) for the existence of such a bounded modulation to a lattice which requires that the window tiles the orthogonal space. Using a special procedure we extend our proof to more general, albeit still non generic, situations including tilings of the Penrose type.