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