|
|
Sfb 288 Differential Geometry and Quantum Physics |
Abstract for Sfb Preprint No. 340
Projection Quasicrystals I: Toral Rotations A. Forrest, J. Hunton and J. Kellendonk
We present a systematic treatment of the commutative and non-commutative topology of quasicrystal point patterns and tilings produced in finite dimensional Euclidean space by the projection or strip method. With no conditions on the projection plane and with a general acceptance domain only weakly constrained, we examine two sets of points constructed by the projection method, one a finite decoration of the other, and their corresponding dynamical systems. We define a projection method pattern or tiling as one whose dynamical system is intermediate to these two systems, concluding that for fixed projection data this allows only a finite number of possible patterns up to topological conjugacy. In all cases the dynamical systems associated to the pattern are almost 1:1 extensions of minimal rotation actions on a torus and we compute these factors explicitly. We establish equivalence between the tiling groupoid and the transformation groupoid of these dynamical systems. In this way, we generalize results of Robinson and of Le and place them in a wider context. The results here provide the necessary groundwork for our second paper in this series, which describes qualitatively the cohomology of projection quasicrystals.
Get a gzip-compressed PostScript copy of this preprint
preprint340.ps.gz (158 kB)