Computation of the gap-labelling for quasi-crystals. (Calcul du label des gaps pour les quasi-cristaux.) (French) Zbl 0996.19006
Summary: We give a proof of the Bellissard gap-labelling conjecture [J. Bellissard, in: From number theory to physics, 538-630 (1992; Zbl 0833.47056)] for quasicrystals. Our main tools are the measured index theorem for laminations on the one hand, and the naturality of the longitudinal Chern character on the other hand.
MSC:
| 19L10 | Riemann-Roch theorems, Chern characters |
Keywords:
gap-labelling conjecture; quasicrystals; measured index theorem for laminations; longitudinal Chern characterCitations:
Zbl 0833.47056References:
| [1] | Bellissard, J., Gap labelling theorem’s for Schrödinger’s operators, (Waldschmidt, M.; Moussa, P.; Luck, J. M.; Itzykson, C., From Number Theory to Physics (1992), Springer), 538-630 · Zbl 0833.47056 |
| [2] | Bellissard, J.; Contensou, E.; Legrand, A., \(K\)-théorie des quasicristaux, image par la trace : le cas du réseau octogonal, C. R. Acad. Sci. Paris, Série I, 327, 197-200 (1998) · Zbl 0920.19003 |
| [3] | M.T. Benameur, H. Oyono-Oyono, Computation of the gap-labelling for quasi-crystal: a foliation approach, Preprint 2001; M.T. Benameur, H. Oyono-Oyono, Computation of the gap-labelling for quasi-crystal: a foliation approach, Preprint 2001 · Zbl 0996.19006 |
| [4] | Connes, A., Noncommutative Geometry (1994), Academic Press: Academic Press New York · Zbl 0681.55004 |
| [5] | Connes, A.; Skandalis, G., The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci., 20, 1139-1183 (1984) · Zbl 0575.58030 |
| [6] | Forrest, A.; Hunton, J., The cohomology and \(K\)-theory of commuting homeomorphisms of the Cantor set, Ergodic Theory Dynamical Systems, 19, 611-625 (1999) · Zbl 0954.54020 |
| [7] | Moore, C. C.; Schochet, C., Global Analysis on Foliated Spaces (1988), Springer: Springer Berlin · Zbl 0648.58034 |
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