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