Gap Labeling

Document Type



This joint review covers:

Moulay-Tahar Benameur and Hervé Oyono-Oyono, Gap-labelling for quasi-crystals (proving a conjecture by J. Bellissard). (English summary) Operator algebras and mathematical physics, (Constanţa, 2001), 11-22, Theta, Bucharest, 2003.

Jerome Kaminker and Ian Putnam, A proof of the gap labeling conjecture, Michigan Mathematical Journal 51(3) (2003), 537-546.

Jean Bellisard, Riccardo Benedetti and Jean-Marc Gambaudo, Spaces of tilings, finite telescopic approximations, and gap-labeling, Communications in Mathematical Physics 261(1) (2006), 1-41.

The gap labeling theorem was originally conjectured by Bellissard [in From number theory to physics (Les Houches, 1989), 538-630, Springer, Berlin, 1992; MR1221111 (94e:46120)]. The problem arises in a mathematical version of solid state physics in the context of aperiodic tilings. Its three proofs, discovered independently by the authors above, all lie in K-theory. Here is the core result of these papers:

Let Σ be a Cantor set and let Σ × ℤd → Σ be a free and minimal action of ℤd on Σ with invariant probability measure μ. Let μ:C(Σ) → ℂ and τμ:C(Σ) ⋊ ℤd → ℂ be the traces induced by μ and denote likewise their induced maps in K-theory. Then μ(K0(C(Σ))) = τμ(K0(C(Σ) × ℤd)) as subsets of ℝ.

We shall try to explain why this core result has anything to do with something called gap labeling.



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