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.

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