6 A. FORREST, J. HUNTON AND J. KELLENDONK
infinitely generated cohomology and also allows for local matching
rules in the sense of [Le]. Furthermore, all projection method patterns
which are used to model quasicrystals seem to have a finitely generated
Ko-gronp. We cannot offer yet an interpretation of the fact that some
patterns produce only finitely many generators for their cohomology
whereas others do not, but, if understood, we hope it could lead to
a criterion to single out a subset of patterns relevant for quasicrystal
physics from the vast set of patterns which may be obtained from the
projection method.
We have mentioned above the motivation from physics to study
the topological theory of point sets or tilings. The theory is also
also of great interest for the theory of topological dynamical sys-
tems, since in d 1 dimensions the dynamical systems mentioned
above have attracted a lot of attention. In [GPS] the meaning of the
non-commutative invariants for the one dimensional case has been
analysed in full detail. Furthermore, substitution tilings give rise to
hyperbolic Z-actions with expanding attractors (hyperbolic attractors
whose topological dimensions are that of their expanding direction)
[AP] a subject of great interest followed up recently by Williams [W]
who conjectures that continuous hulls of substitution tilings (called
tiling spaces in [W]) are fiber bundles over tori with the Cantor set
as fiber. We have not put emphasis on this question but it may be
easily concluded from our analysis of Section 1.10 that the continuous
hulls of projection method tilings are always Cantor set fiber bundles
over tori (although these tilings are rarely substitutional and there-
fore carry in general no obvious hyperbolic Z-action). Anderson and
Putnam [AP] and one of the authors [K2] have employed the substi-
tution of the tiling to calculate topological invariants of it.
Organization of the book The order of material in this memoir is
as follows. In Chapter I we define and describe the various dynamical
systems mentioned above, and examine their topological relationships.
These are compared with the pattern groupoid and its associated C*
algebra in Chapter II where these latter objects are introduced. Also
in Chapter II we set up and prove the equivalence of our various
topological invariants; we end the chapter by demonstrating how these
invariants provide an obstruction to a pattern being self-similar.
The remaining chapters offer three illustrations of the eomputab-
ility of these invariants. In Chapter III we give a complete calculation
for all 'codimension one' projection patterns - patterns arising from
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