computed explicitly. This gives a clear picture of the orbits of non-
singular points in the pattern dynamical system. A precursor to our
description of the pattern dynamical system can be found in the work
of Robinson [R2], who examined the dynamical system of the Pen-
rose tiling and showed that it is an almost 1-1 extension of a minimal
action by rotation on a 4-torus. Although Robinson used quite
special properties of the tiling, Hof [H] has noted that these tech-
niques are generalizable without being specific about the extent of
the generalization.
Our approach is quite different from that of Robinson and, by
constructing a larger topological space from which the pattern dy-
namical system is formed by a quotient, we follow most closely the
approach pioneered by Le [Le] as noted in the General Introduction.
The care taken here in the topological foundations seems necessary for
further progress and to allow general acceptance domains. Even in the
canonical case, Corollary 7.2 and Proposition 8.4 of this chapter, for
example, require this precision despite being direct generalizations of
Theorem 3.8 in [Le]. Also, as mentioned in the General Introduction,
many of the results of this Chapter are to be found independently in
2 The projection method and associated geometric
We use the construction of point patterns and tilings given in Chapters
2 and 5 of Senechal's monograph [Se] throughout this paper, adding
some assumptions on the acceptance domain in the following defini-
Definitions 2.1 Consider the lattice
sitting in standard posi-
tion inside
(i.e. it is generated by an orthonormal basis of
Suppose that E is a d dimensional subspace of R ^ and
its or-
thocomplement. For the time being we shall make no assumptions
about the position of either of these planes.
Let 7 T be the projection onto E and
the projection onto E^.
Let Q E +
(Euclidean closure). This is a closed subgroup
Let K be a compact subset of
which is the closure if its
interior (which we write IntK) in
Thus the boundary of K in

is compact and nowhere dense. Let S = K + E, a subset of
sometimes refered to as the strip with acceptance domain K.
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