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, (MPu,Rd) and (MPu,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 MPU — MPU, 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