The torus parametrization of quasiperiodic LI-classes

Abstract

The torus parametrization of quasiperiodic local isomorphism classes is introduced and used to determine the number of elements in such a class with special symmetries or inflation properties. The method is explained in an illustrative fashion for some widely used tiling classes with golden mean rescaling, namely for the Fibonacci chain (one-dimensional), the triangle and Penrose patterns (two-dimensional) and for Kramer's and Danzer's icosahedral tilings (three-dimensional). We obtain a rather complete picture of the orbit structure within these classes, and also discuss various general results.