The Local Structure of Tilings and their Integer Group of Coinvariants

Details The Local Structure of Tilings and their Integer Group of Coinvariants The Local Structure of Tilings and their Integer Group of Coinvariants

1995-08-02

arxiv.org/abs/cond-mat/9508010v1

Commun.Math.Phys. 187 (1997) 115-158

Physics

Condensed Matter

45 pages including 9 figures, LaTex

Scientific paper

10.1007/s002200050131

The local structure of a tiling is described in terms of a multiplicative structure on its pattern classes. The groupoid associated to the tiling is derived from this structure and its integer group of coinvariants is defined. This group furnishes part of the $K_0$-group of the groupoid $C^*$-algebra for tilings which reduce to decorations of $\Z^d$. The group itself as well as the image of its state is computed for substitution tilings in case the substitution is locally invertible and primitive. This yields in particular the set of possible gap labels predicted by $K$-theory for Schr\"odinger operators describing the particle motion in such a tiling.

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