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.
Readership
Graduate students and research mathematicians
interested in convex and discrete geometry.