TABLE OF CONTENTS
General Introduction 1
I Topological Spaces and Dynamical Systems 10
1 Introduction 10
2 The projection method and associated geometric
constructions 11
3 Topological spaces for point patterns 16
4 Tilings and point patterns 20
5 Comparing 11^ jand n
n
24
6 Calculating MPU and MPu 26
7 Comparing MPU with MPU 30
8 Examples and counter-examples 33
9 The topology of the continuous hull 37
10 A Cantor
Zd
dynamical system 40
II Groupoids, C*-algebras, and their Invariants 46
1 Introduction 46
2 Equivalence of projection method pattern groupoids 47
3 Continuous similarity of projection method pattern
groupoids 54
4 Pattern cohomology and If-theory 58
5 Homological conditions for self similarity 61
III Approaches to Calculation I: Cohomology for
Codimension One 64
1 Introduction 64
2 Inverse limit acceptance domains 64
3 Cohomology in the case d = N 1 66
IV Approaches to Calculation II: Infinitely Generated
Cohomology 69
1 Introduction 69
2 The canonical projection tiling 69
3 Constructing C-topes 74
4 The indecomposable case 80
5 The decomposable case 86
6 Conditions for infinitely generated cohomology 89
V Approaches to Calculation III: Cohomology for
Small Codimension 94
1 Introduction 94
2 Set up and statement of the results 95
3 Complexes defined by the singular spaces 99
vn
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