the non-commutative topology of the C*-algebra of observables and
the topological invariants of T are the invariants of AT- In particu-
lar, we shall be interested in the K-theory of AT- Its i^o-group has
direct relevance to physics through the gap-labelling. In fact, Jean
Bellissard's K-theoretic formulation of the gap-labelling stands at the
beginning of this approach [Bl].
The second way of looking at the topological invariants of a dis-
crete point set T begins with the construction of the continuous hull
of T. There are various ways of defining a metric on a set of patterns
through comparison of their local configurations. Broadly speaking,
two patterns are deemed close if they coincide on a large window
around the origin 0 E R
up to a small discrepancy. It is the way
the allowed discrepancy is quantified which leads to different metric
topologies and we choose here one which has the strongest compact-
ness properties, though we have no intrinsic motivation for this. The
continuous hull of T is the closure, MT', of the set of translates of
T with respect to this metric. We use the notation MT because it
is essentially a mapping torus construction for a generalized discrete
dynamical system:
acts on MT by translation and the set fir of
all elements of MT which (as point sets) contain 0 forms an abstract
transversal called the discrete hull. If d were 1 then fir would give
rise to a Poincare section, the intersection points of the flow line of
the action of R with fir defining an orbit of a Z action in fir, and
A4T would be the mapping torus of that discrete dynamical system
(fir,Z). For larger d one cannot expect to get a
action on fir
in a similar way but finds instead a generalized discrete dynamical
system which can be summarized in a groupoid QT whose unit space
is the discrete hull fir- Topological invariants for T are therefore the
topological invariants of MT and of QT and we shall be interested in
particular in their cohomologies. We define the cohomology of T to be
that of QT. Under a finite type condition, namely that for any given
r there are only a finite number of translational congruence classes
of subsets which fit inside a window of diameter r, the algebra AT
sketched above is isomorphic to the groupoid C*-algebra of QT. This
links the two approaches.
Having outlined the general philosophy we hasten to remark that
we will not explain all its aspects in the main text. In particular, we
have nothing to say there about the physical aspects of the theory and
the description of the algebra of observables, referring here the reader
to [BZH] [KePu], or to the more original literature [Bl] [B2] [Kl].
Instead, our aim in this memoir is to discuss and compare the different
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