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Item Details
| Title:
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TOPOLOGICAL INVARIANTS FOR PROJECTION METHOD PATTERNS
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| By: |
Alan Forrest, John Hunton, Johannes Kellendonk |
| Format: |
Paperback |
| List price:
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£58.00 |
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| ISBN 10: |
0821829653 |
| ISBN 13: |
9780821829653 |
| Publisher: |
AMERICAN MATHEMATICAL SOCIETY |
| Pub. date: |
15 August, 2002 |
| Series: |
Memoirs of the American Mathematical Society No. 159 |
| Pages: |
120 |
| Description: |
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. |
| Synopsis: |
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, formulae 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. |
| Illustrations: |
bibliography |
| Publication: |
US |
| Imprint: |
American Mathematical Society |
| Returns: |
Returnable |
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