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 MP U 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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