TOPOLOGICAL SPACES AND DYNAMICAL SYSTEMS 11

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

R2

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

[Sch].

2 The projection method and associated geometric

constructions

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-

tions.

Definitions 2.1 Consider the lattice

ZN

sitting in standard posi-

tion inside

RN

(i.e. it is generated by an orthonormal basis of

WN).

Suppose that E is a d dimensional subspace of R ^ and

E1-

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

TT1-

the projection onto E^.

Let Q — E +

ZN

(Euclidean closure). This is a closed subgroup

oiRN.

Let K be a compact subset of

E1-

which is the closure if its

interior (which we write IntK) in

E1-.

Thus the boundary of K in

E±

is compact and nowhere dense. Let S = K + E, a subset of

RN

sometimes refered to as the strip with acceptance domain K.