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
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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