8 A. FORREST, J. HUNTON AND J. KELLENDONK mention the historical developments in the introduction to Chapter I. Our approach constructs a large topological space from which the pattern dynamical system is formed by a quotient and so we follow most closely the idea pioneered by Le [Le] for the case of canonical projection tilings. The "Cantorization" of Euclidean space by corners or cuts, as described by Le and others (see [Le] [H] etal.), is produced in our general topological context in sections 1.3, 1.4 and 1.9. In this, we share the ground with Schlottmann [Sch] and Herrmann [He] who have recently established the results of Chapter I in such (and even greater) generality, Schlottmann in order to generalize results of Hof and describe the unique ergodicity of the underlying dynamical systems and Herrmann to draw a connection between codimension 1 projection patterns and Denjoy homeomorphisms of the circle. We mention this relation at the end of chapter III. Bellissard, Contensou and Legrand [BCL] compare the C*- algebra of a dynamical groupoid with a C*-algebra of operators defined on a class of tilings obtained by projection, the general theme of Chapter II. Using a Rosenberg Shochet spectral sequence, they also establish, for 2-dimensional canonical projection tilings, an equation of dynamical cohomology and C* if-theory in that case. It is the first algebraic topological approach to projection method tiling if-theory found in the literature. We note, however, that the groupoid they con- sider is not always the same as the tiling groupoid we consider, nor do the dynamical systems always agree the Penrose tiling is a case in point, where we find that KQ of the spaces considered in [BCL] is Z°°. The difference may be found in the fact that we consider a given projection method tiling or pattern and its translates, while they con- sider a larger set of tilings, two elements of which may sometimes be unrelated by approximation and translation parallel to the projection plane. Acknowledgements. We thank F. Gahler for verifying and assisting our results (V.6) on the computer and for various useful comments and M. Baake for reading parts of the manuscript. The third au- thor would like to thank J. Bellissard for numerous discussions and constant support. The collaboration of the first two authors was initiated by the William Gordon Seggie Brown Fellowship at The University of Ed- inburgh, Scotland, and received continuing support from a Collab- orative Travel Grant from the British Council and the Research Council of Norway with the generous assistance of The University
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