I Topological Spaces and Dynamical Systems 1 Introduction In this chapter our broad goal is to study the topology and associ- ated dynamics of projection method patterns, while imposing only few restrictions on the freedom of the construction. From a specific set of projection data we define and examine a number of spaces and dynamical systems and their relationships from these constructions, in later chapters, we set up our invariants, defined via various topo- logical and dynamical cohomology theories. In Chapter II we shall also compare the commutative spaces of this chapter with the non- commutative spaces considered by other authors in e.g. [B2] [BCL] [AP] [K2]. Given a subspace, E, acceptance domain, K, and a position- ing parameter u, we distinguish two particular Rd dynamical systems constructed by the projection method, (MP u ,Rd) and (MP u ,Md), the first automatically a factor of the second. This allows us to define a projection method pattern (with data (E,K,u)) as a pattern, T, whose dynamical system, (MT,R d ), is intermediate to these two ex- treme systems. Sections 3 to 9 of this chapter provide a complete description of the spaces and the extension MP U — MP U , showing, under further weak assumptions on the acceptance domain, that it is a finite isometric extension. In section 7 we conclude that this re- stricts (MT, ]Rrf) to one of a finite number of possibilities, and that any projection method pattern is a finite decoration of its correspond- ing point pattern Pu (2.1). The essential definitions are to be found in Sections 2 and 4. In section 10 we describe yet another dynamical system connected with a projection method pattern, this time a Zd action on a Cantor set X, whose mapping torus is the space of the pattern dynamical system. It will be this dynamical system that, in chapters 3, 4 and 5, will allow the easiest computation and discussion of the behaviour of our invariants. For the canonical case with E±nZN = 0 this is the same system as that constructed in [BCL]. All the dynamical systems produced in this memoir are almost 1-1 extensions of an action of Rd or Zd by rotations (Def. 2.15) on a torus (or torus extended by a finite abelian group). In each case the dimension of the torus and the generators of the action can be 10

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