Skip to main content
SpringerLink
Log in

Spaces of Tilings, Finite Telescopic Approximations and Gap-Labeling

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The continuous Hull of a repetitive tiling T in ℝd with the Finite Pattern Condition (FPC) inherits a minimal ℝd-lamination structure with flat leaves and a transversal which is a Cantor set. This class of tiling includes the Penrose & the Amman Benkker ones in 2D, as well as the icosahedral tilings in 3D. We show that the continuous Hull, with its canonical ℝd-action, can be seen as the projective limit of a suitable sequence of branched, oriented and flat compact d-manifolds. As a consequence, the longitudinal cohomology and the K-theory of the corresponding C*-algebra are obtained as direct limits of cohomology and K-theory of ordinary manifolds. Moreover, the space of invariant finite positive measures can be identified with a cone in the d th homology group canonically associated with the orientation of ℝd. At last, the gap labeling theorem holds: given an invariant ergodic probability measure μ on the Hull the corresponding Integrated Density of States (IDS) of any selfadjoint operators affiliated to takes on values on spectral gaps in the ℤ-module generated by the occurrence probabilities of finite patches in the tiling.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anderson, J.E., Putnam, I.: Topological invariants for substitution tilings and their associated C*-algebra. Erg. Th. Dynam. Syst. 18, 509–537 (1998)

    Article  MATH  Google Scholar 

  2. Atiyah, M. F.: K-Theory, Harvard University Lecture Notes, 1964

  3. Bellissard, J.: Schrödinger's operators with an almost periodic potential: an overview, Lecture Notes in Phys., n°153, Berlin-Heidelberg-NewYork, Springer Verlag, 1982, pp. 356–363

  4. Bellissard, J.: K-Theory of C*-algebras in Solid State Physics. In: Statistical Mechanics and Field Theory, Mathematical Aspects, T.C. Dorlas, M.N. Hugenholtz, M. Winnink, eds., Lecture Notes in Physics, n°257, Berlin-Heidelberg-NewYork, Springer, 1986, pp. 99-156

  5. Bellissard, J.: Gap labeling Theorems for Schrödinger's Operators. In From Number Theory to Physics, Les Houches March 89, J.M. Luck, P. Moussa, M. Waldschmidt eds., Berlin-Heidelberg-NewYork, Springer, (1993), pp.538-630

  6. Bellissard, J., Bovier, A. Ghez, J.-M.: Gap labeling Theorems for One Dimensional Discrete, Schrödinger Operators. Rev. Math. Phys. 4, 1–37 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bellissard, J., Contensou, E., Legrand, A.: K-théorie des quasi-cristaux, image par la trace: le cas du réseau octogonal. C. R. Acad. Sci. (Paris), t.327, Série I, 197-200 (1998)

  8. Bellissard, J., Hermmann, D., Zarrouati, M.: Hull of Aperiodic Solids and Gap labeling Theorems. In: Directions in Mathematical Quasicrystals, CRM Monograph Series, Volume 13, M.B. Baake, R.V. Moody, eds., Providence, RI: AMS, 2000, pp. 207–259

  9. Bellissard J., Kellendonk, J., Legrand, A.: Gap-labeling for three dimensional aperiodic solids. C. R. Acad. Sci. (Paris), t.332, Série I, 521–525 (2001)

  10. Bellissard, J.: Noncommutative Geometry of Aperiodic Solids. In: Proceedings of the 2001 Summer School of Theoretical Physics, Geometry, Topology and Quantum Field Theory, Villa de Leyva, Colombia, 7–30 July 2001, River Edge NJ: World Scientific, 2003

  11. Benameur, M., Oyono, H.: Private communication.

  12. Benedetti, R., Gambaudo, J.-M.: On the dynamics of -solenoids. Applications to Delone sets. Ergod. Th. & Dynam. Sys. 23, 673–691 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Benedetti, R., Petronio, C.: Lectures on Hyperbolic Geometry, Berlin-Heidelberg-New York: Springer,1992

  14. Benedetti, R., Petronio, C.: Branched Standard Spines of 3-Manifolds. LNM 1653, Berlin-Heidelberg-New York: Springer, 1997

  15. Blackadar, B.: K-Theory for Operator Algebras. MSRI Publications, Vol.5, Berlin-Heidelberg-New York: Springer-Verlag, 1986

  16. Connes, A.: Sur la théorie non commutative de l'intégration. In: Algèbres d'Opérateurs, Lecture Notes in Mathematics, 725, Berlin: Springer, Berlin 1979 pp. 19–143

  17. Connes, A.: ``An analogue of the Thom isomorphism for crossed products of a C*-algebra by an action of R''. Adv. in Math. 39, 31–55 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  18. Connes, A.: Cyclic cohomology and the transverse fundamental class of a foliation. Pitman Res. Notes in Math. 123, London: Longman Harlow, 1986, pp. 52–144

  19. Connes, A.: Noncommutative Geometry. San Diego: Academic Press, 1994

  20. Ghys, E.: Laminations par surfaces de Riemann. In: Dynamique et géométrie complexes, Panoramas & Synthèses 8, 49–95 (1999)

    MATH  MathSciNet  Google Scholar 

  21. Forrest, A., Hunton, J.: The cohomology and K-theory of commuting homeomorphisms of the Cantor set. Erg. Th. Dynam. Syst. 19, 611–625 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Forrest, A., Hunton, J., Kellendonk, J.: Cohomology of canonical projection tilings. Commun. Math. Phys. 226, 289–322 (2002); Topological invariants for projection point patterns. Memoirs of the Amer. Math. Soc. Vol I, 159/758 providence RI Amer. Math. Soc., 2002; Cohomology groups for projection point patterns. In: Proceedings of the International Congress of Mathematical Physics 2000, Boston MA: Int press 2001, pp. 333–339

    Article  ADS  MATH  MathSciNet  Google Scholar 

  23. van Elst, A.: Gap labeling Theorems for Schrödinger's Operators on the Square and Cubic Lattice. Rev. Math. Phys. 6, 319–342 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  24. Gähler F., Kellendonk, J.: Cohomology groups for projection tilings of codimension 2. Mat. Sci. Eng. A 294–296, 438–440 (2000)

    Google Scholar 

  25. Gambaudo, J.-M., Martens, M.: Algebraic topology for minimal Cantor sets. Preprint, Dijon (2000), at http://www.dim.uchile. cl/gambaudo/GM.pdf

  26. Gottschalk, W., Hedlund, G.A.: Topological Dynamics. providence RI: Amer. Math. Soc., 1955

  27. Griffiths, P, Harris, J.: Principles of Algebraic Geometry. New York: Wiley 1978

  28. Hippert, F., Gratias, D. eds.: Lectures on Quasicrystals. Les Ulis: Editions de Physique, 1994

  29. Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84 403–438 (1982)

    Google Scholar 

  30. Kaminker, J., Putnam, I.F.: Private communication.

  31. Kellendonk, J., Putnam, I.F.: Tilings, C*-algebras and K-theory. In: Directions in Mathematical Quasicrystals, CRM Monograph Series, Volume 13, M.P. Baake, R.V. Moody, eds., Providence, RI: AMS, 2000, pp. 177–206

  32. Lagarias, J.C., Pleasants, P.A.B.: Repetitive Delone sets and quasicrystals. Ergod. Th. & Dynam. Sys. 23, 831–867 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  33. Milnor, J.W., Stasheff, J.D.: Characteristic Classes. Princeton, NJ: Princeton University Press, 1974

  34. Moore, C.C., Schochet, C.: Global Analysis on Foliated Spaces. Math. Sci. Res. Inst. Publ. No. 9, New York: Springer-Verlag, 1988

  35. Pedersen, G.: C* -algebras. and their Automorphism Group. London: Academic Press, 1979

  36. Pimsner, M.V., Voiculescu, D.: Exact sequences for K-groups and ext-groups of certain cross-product of C*-algebra. J. Operator Theory 4, 93–118 (1980)

    MATH  MathSciNet  Google Scholar 

  37. Pimsner, M.V.: Range of traces on K 0 of reduced crossed products by free groups. In: Operator Algebras and their Connections with Topology and Ergodic Theory, Lecture Notes in Math., n°1132, Berlin-Heidelberg-New York: Springer Verlag, 1983, pp. 374–408

  38. Putnam, I.F.: The C ★ algebras associated with minimal homeomorphisms of the Cantor set. Pac. J. Math. 136, 329–352 (1989)

    MATH  MathSciNet  Google Scholar 

  39. Radin, C.: Miles of Tiles. Student Mathematical Library, Vol 1 providence RI: Amer. Math. Soc., 1999; see also Radin, C.: The pinwheel tilings of the plane. Ann. Math. 139, 661–702 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  40. Sadun, L., Williams, R.F.: Tiling spaces are Cantor fiber bundles Ergith and Dya. Systs. 23, 307–316(2003)

    MATH  MathSciNet  Google Scholar 

  41. Senechal, M.: Crystalline Symmetries : An informal mathematical introduction, Institute of Physics, London: Alan Hilger, Ltd., 1990

  42. Serre, J.-P.: Modules projectifs et espaces fibrés à fibre vectorielle, (1958) Séminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot, 1957/58, Fasc. 2, Exposé 23, Paris: Secrétariat mathématique, 18 pp.

  43. Spanier, E.H.: Algebraic Topology, New York: McGraw-Hill, 1966

  44. Sullivan, D.: Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math. 89, 225–255 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  45. Swan, R.G.: Vector bundles and projective modules. Trans. Amer. Math. Soc. 105, 264–277 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  46. Thurston, W.: Geometry and Topology of 3-manifolds. princeton, NJ: Princeton University Press 1979

  47. Versik, A.M.: A Theorem on Periodical Markov Approximation in Ergodic Theory. In: Ergodic Theory and Related Topics (Vitte, 1981), Math. Res., 12, Berlin: Akademie-Verlag, 1981, pp. 195–206

  48. Williams, R.F.: Expanding attractors. Publ. IHES 43 169–203 (1974)

Download references

Author information

Authors and Affiliations

  1. Georgia Institute of Technology, School of Mathematics, Atlanta, GA, 30332-0160, USA

    Jean Bellissard

  2. Institut Universitaire de France,  

    Jean Bellissard

  3. Dipartimento di Matematica,  , via F. Buonarroti 2, 56127, Pisa, Italy

    Riccardo Benedetti

  4. Centro de Modelamiento Matematico, UMI CNRS 2807, Universidad de Chile, Blanco Encalada, 2120, Santiago, Chile

    Jean-Marc Gambaudo

Authors
  1. Jean Bellissard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Riccardo Benedetti
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jean-Marc Gambaudo
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Connes

Accepted in Revised Form: 7 May 2005

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bellissard, J., Benedetti, R. & Gambaudo, JM. Spaces of Tilings, Finite Telescopic Approximations and Gap-Labeling. Commun. Math. Phys. 261, 1–41 (2006). https://doi.org/10.1007/s00220-005-1445-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-005-1445-z

Keywords

Search

Navigation