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Memoirs of the American Mathematical Society
2002; 120 pp; softcover
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Order Code: MEMO/159/758
This memoir develops, discusses and compares a range of commutative and non-commutative invariants defined for projection method tilings and point patterns. The projection method refers to patterns, particularly the quasiperiodic patterns, constructed by the projection of a strip of a high dimensional integer lattice to a smaller dimensional Euclidean space. In the first half of the memoir the acceptance domain is very general -- any compact set which is the closure of its interior -- while in the second half we concentrate on the so-called canonical patterns. The topological invariants used are various forms of \(K\)-theory and cohomology applied to a variety of both \(C^*\)-algebras and dynamical systems derived from such a pattern.
The invariants considered all aim to capture geometric properties of the original patterns, such as quasiperiodicity or self-similarity, but one of the main motivations is also to provide an accessible approach to the the \(K_0\) group of the algebra of observables associated to a quasicrystal with atoms arranged on such a pattern.
The main results provide complete descriptions of the (unordered) \(K\)-theory and cohomology of codimension 1 projection patterns, formulæ for these invariants for codimension 2 and 3 canonical projection patterns, general methods for higher codimension patterns and a closed formula for the Euler characteristic of arbitrary canonical projection patterns. Computations are made for the Ammann-Kramer tiling. Also included are qualitative descriptions of these invariants for generic canonical projection patterns. Further results include an obstruction to a tiling arising as a substitution and an obstruction to a substitution pattern arising as a projection. One corollary is that, generically, projection patterns cannot be derived via substitution systems.
Graduate students and research mathematicians interested in convex and discrete geometry.
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