Communications in Mathematical Physics

, Volume 255, Issue 3, pp 655–681 | Cite as

Quasicrystals and Almost Periodicity

Article

Abstract

We give in this paper topological and dynamical characterizations of mathematical quasicrystals. Let Open image in new window denote the space of uniformly discrete subsets of the Euclidean space. Let Open image in new window denote the elements of Open image in new window that admit an autocorrelation measure. A Patterson set is an element of Open image in new window such that the Fourier transform of its autocorrelation measure is discrete. Patterson sets are mathematical idealizations of quasicrystals. We prove that SOpen image in new window is a Patterson set if and only if S is almost periodic in (Open image in new window,Open image in new window), where Open image in new window denotes the Besicovitch topology. Let χ be an ergodic random element of Open image in new window. We prove that χ is almost surely a Patterson set if and only if the dynamical system has a discrete spectrum. As an illustration, we study deformed model sets.

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References

  1. 1.
    Baake, M., Lenz, D.: Deformation of delone dynamical systems and pure point diffraction. To appear in Journal of Fourie Analysis and ApplicationsGoogle Scholar
  2. 2.
    Baake, M., Moody, R.V. (eds.): Directions in mathematical quasicrystals, Volume 13 of CRM Monograph Series. Providence, RI: American Mathematical Society, 2000Google Scholar
  3. 3.
    Baake, M., Moody, R.V.: Weighted Dirac combs with pure point diffraction. J. für die Reine and Angewandte Math. In pressGoogle Scholar
  4. 4.
    Baake, M., Moody,R.V., Pleasants, P.A.B.: Diffraction from visible lattice points and kth power free integers. Selected papers in honor of Ludwig Danzer, Discrete Math. 221(1–3), 3–42 (2000)Google Scholar
  5. 5.
    Bernuau, G., Duneau, M.: Fourier analysis of deformed model sets. In: Directions in mathematical quasicrystals, Providence, RI: Am. Math. Soc. 2000, pp. 43–60Google Scholar
  6. 6.
    Besicovitch, A.S., Bohr, H.: Almost periodicity and general trigonometric series. Acta Math. 57, 203–292 (1931)Google Scholar
  7. 7.
    Dworkin, S.: Spectral theory and x-ray diffraction. J. Math. Phys. 34(7), 2965–2967 (1993)Google Scholar
  8. 8.
    Eberlein, W.F.: A note on Fourier-Stieltjes transforms. Proc. Am. Math. Soc. 6, 310–312, (1955)Google Scholar
  9. 9.
    de Lamadrid, G.J., Argabright, L.N.: Almost periodic measures. Mem. Am. Math. Soc. 85(428):vi+219 (1990)Google Scholar
  10. 10.
    Gouéré, J.B.: Diffraction et mesure de Palm des processus ponctuels. C. R. Math. Acad. Sci. Paris 336(1), 57–62 (2003)Google Scholar
  11. 11.
    Hewitt, E.: Representation of functions as absolutely convergent Fourier-Stieltjes transforms. Proc. Am. Math. Soc. 4, 663–670 (1953)Google Scholar
  12. 12.
    Hof, A.: On diffraction by aperiodic structures. Commun. Math. Phys. 169(1), 25–43 (1995)Google Scholar
  13. 13.
    Katz, A.: Introduction aux quasicristaux. Séminaire Bourbaki. Vol. 1997/98, Astérisque (252):Exp. No. 838(3), 81–103 (1998)Google Scholar
  14. 14.
    Katznelson, Y.: An introduction to harmonic analysis. Corrected edition, New York: Dover Publications Inc., 1976Google Scholar
  15. 15.
    Lagarias, J.C.: Mathematical quasicrystals and the problem of diffraction. In: Directions in mathematical quasicrystals, Volume 13 of CRM Monogr. Ser., Providence, RI: Am. Math. Soc., 2000, pp. 61–93Google Scholar
  16. 16.
    Lagarias, J.C., Pleasants, P.A.B.: Repetitive Delone sets and quasicrystals. Ergodic Theory Dynam. Systems 23(3), 831–867 (2003)Google Scholar
  17. 17.
    Lee, J.-Y., Moody, R.V., Solomyak, B.: Pure point dynamical and diffraction spectra. Ann. Henri Poincaré 3(5), 1003–1018 (2002)Google Scholar
  18. 18.
    Lee, J.-Y., Moody, R.V., Solomyak, B.: Consequences of pure point diffraction spectra for multiset substitution systems. Discrete Comput. Geom. 29(4), 525–560 (2003)Google Scholar
  19. 19.
    Matheron, G.: Random sets and integral geometry. With a foreword by Geoffrey S. Watson, Wiley Series in Probability and Mathematical Statistics, New York-London-Sydney: John Wiley & Sons, 1975Google Scholar
  20. 20.
    Møller, J.: Lectures on random Voronoi tessellations. New York: Springer-Verlag, 1994Google Scholar
  21. 21.
    Moody, R.V. (ed.): The mathematics of long-range aperiodic order, Vol. 489 of NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Dordrecht: Kluwer Academic Publishers Group, 1997Google Scholar
  22. 22.
    Moody, R.V.: Uniform distribution in model sets. Canad. Math. Bull. 45(1), 123–130 (2002)Google Scholar
  23. 23.
    Moody, R.V.: Mathematical quasicrystals: A tale of two topologies. To appear in the Proceedings of the International Congress of Mathematical Physics, Lisbon, 2003, World Scientific Publishing CompanyGoogle Scholar
  24. 24.
    Moody, R.V., Strungaru, N.: Point sets and dynamical systems in the autocorrelation topology. Canad. Math. Bull. 47(1), 82–99 (2004)Google Scholar
  25. 25.
    Neveu, J.: Processus ponctuels. In: École d’Été de Probabilités de Saint-Flour, VI—1976, Lecture Notes in Math., Vol. 598. Berlin: Springer-Verlag, 1977, pp. 249–445Google Scholar
  26. 26.
    Queffélec, M.: Substitution dynamical systems–-spectral analysis, Volume 1294 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1987Google Scholar
  27. 27.
    Reed, M., Simon, B.: Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1975Google Scholar
  28. 28.
    Rudin, W.: Real and complex analysis: Third edition, New York: McGraw-Hill Book Co., 1987Google Scholar
  29. 29.
    Rudin, W.: Fourier analysis on groups. Reprint of the 1962 original, A Wiley-Interscience Publication, New York: John Wiley & Sons Inc., 1990Google Scholar
  30. 30.
    Schlottmann, M.: Generalized model sets and dynamical systems. In: Directions in mathematical quasicrystals, Providence, RI: Am. Math. Soc., 2000, pp. 143–159Google Scholar
  31. 31.
    Segal, I.E.: The class of functions which are absolutely convergent Fourier transforms. Acta Sci. Math. Szeged 12(Leopoldo Fejer et Frederico Riesz LXX annos natis dedicatus, Pars B) 157–161 (1950)Google Scholar
  32. 32.
    Senechal, M.: Quasicrystals and geometry. Cambridge: Cambridge University Press, 1995Google Scholar
  33. 33.
    Shechtman, D., Blech, I., Gratias, D., Cahn, J.W.: Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953 (1984)CrossRefGoogle Scholar
  34. 34.
    Solomyak, B.: Dynamics of self-similar tilings. Ergodic Theory Dynam. Systems 17(3), 695–738 (1997)Google Scholar
  35. 35.
    Solomyak, B.: Spectrum of dynamical systems arising from Delone sets. In: Quasicrystals and discrete geometry (Toronto, ON, 1995), Providence, RI: Am. Math. Soc., 1998, pp. 265–275Google Scholar
  36. 36.
    Wiener, N.: The ergodic theorem. Duke Math. 5, 1–18 (1939)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.LaPCS, Bâtiment B, Domaine de GerlandUniversité Claude Bernard Lyon 1Lyon Cedex 07France

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