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
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