Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T14:21:02.124Z Has data issue: false hasContentIssue false
  • English
  • Français

Equicontinuous Delone Dynamical Systems

Published online by Cambridge University Press:  20 November 2018

Johannes Kellendonk  and
Daniel Lenz
Johannes Kellendonk
Affiliation:
Université de Lyon, Université Claude Bernard Lyon 1, Institut Camille Jordan, CNRS UMR 5208, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne cedex, France, e-mail: kellendonk@math.univ-lyon1.fr
Daniel Lenz
Affiliation:
Mathematisches Institut, Friedrich-Schiller Universität Jena, Ernst-Abbé Platz 2, D-07743 Jena, Germany, e-mail: daniel.lenz@uni-jena.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We characterize equicontinuous Delone dynamical systems as those coming from Delone sets with strongly almost periodic Dirac combs. Within the class of systems with finite local complexity, the only equicontinuous systems are then shown to be the crystallographic ones. On the other hand, within the class without finite local complexity, we exhibit examples of equicontinuous minimal Delone dynamical systems that are not crystallographic. Our results solve the problem posed by Lagarias as to whether a Delone set whose Dirac comb is strongly almost periodic must be crystallographic.

Keywords

37B50Delone setstilingsdiffractiontopological dynamical systemsalmost periodic systems
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 1 , 01 February 2013 , pp. 149 - 170
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Auslander, J., Minimal flows and their extensions. North-Holland Mathematics Studies, 153, North-Holland Publishing Co., Amsterdam, 1988.Google Scholar
[2] Baake, M. and Lenz, D., Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra. Ergodic Theory Dynam. Systems 24(2004), no. 6, 1867–1893. http://dx.doi.org/10.1017/S0143385704000318 Google Scholar
[3] Baake, M., Deformation of Delone dynamical systems and pure point diffraction. J. Fourier Anal. Appl. 11(2005), no. 2, 125–150. http://dx.doi.org/10.1007/s00041-005-4021-1 Google Scholar
[4] Baake, M.,Lenz, D., and Moody, R. V., Characterization of model sets by dynamical systems. Ergodic Theory Dynam. Systems 27(2007), no. 2, 341–382. http://dx.doi.org/10.1017/S0143385706000800 Google Scholar
[5] Baake, M.,Lenz, D., and Richard, C., Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies. Lett. Math. Phys. 82(2007), no. 1, 61–77. http://dx.doi.org/10.1007/s11005-007-0186-7 Google Scholar
[6] Baake, M. and Moody, R. V. (eds), Directions in mathematical quasicrystals. CRM Monograph Series, 13, American Mathematical Society, Providence, RI, 2000.Google Scholar
[7] Baake, M., Weighted Dirac combs with pure point diffraction. J. Reine Angew. Math. 573(2004), 61–94. http://dx.doi.org/10.1515/crll.2004.064 Google Scholar
[8]Barge, M. and Diamond, B., Proximality in Pisot tiling spaces. Fund. Math. 194(2007), no. 3, 191–238. http://dx.doi.org/10.4064/fm194-3-1 Google Scholar
[9] Barge, M. and Kellendonk, J., Proximality and pure point spectrum for tiling dynamical systems. arxiv:1108.4065.Google Scholar
[10] Barge, M. and Olimb, C., Asymptotic structure in substitution tiling spaces. arxiv:1101.4902.Google Scholar
[11]Barge, M. and Smith, M., Augmented dimension groups and ordered cohomology. Ergodic Theory Dynam. Systems 29(2009), no. 1, 1–35. http://dx.doi.org/10.1017/S0143385708080449 Google Scholar
[12]Bellissard, J., Herrmann, D. J. L., and Zarrouati, M., Hulls of aperiodic solids and gap-labelling theorems. In: Directions in mathematical quasicrystals. CRM Monogr. Ser., 13, American Mathematical Society, Providence, RI, 2000.Google Scholar
[13] Córdoba, A., La formule sommatoire de Poisson. C. R. Acad. Sci. Paris, Sér. I Math. 306(1988), no. 8, 373–376.Google Scholar
[14]Forrest, A. H., Hunton, J. R., and Kellendonk, J., Topological invariants for projection method patterns. Mem. Amer. Math. Soc. 159(2002), no. 758.Google Scholar
[15] Gil de Lamadrid, J. and Argabright, L. N., Almost periodic measures. Mem. Amer. Math. Soc. 85(1990), no. 428.Google Scholar
[16] Gouéré, J.-B., Diffraction et mesure de Palm des processus ponctuels. C. R. Acad. Sci. 336(2003), no. 1, 57–62.Google Scholar
[17] Gouéré, J.-B., Quasicrystals and almost periodicity. Commun. Math. Phys. 255(2005), no. 3, 655–681. http://dx.doi.org/10.1007/s00220-004-1271-8 Google Scholar
[18]Hof, A., On diffraction by aperiodic structures. Commun. Math. Phys. 169(1995), no. 1, 25–43. http://dx.doi.org/10.1007/BF02101595 Google Scholar
[19] Ishimasa, T., Nissen, H.-U., and Fukano, Y., New ordered state between crystallographic and amorphous in Ni-Cr particles. Phys. Rev. Lett. 55(1985), no. 5, 511–513.Google Scholar
[20]Janot, C., Quasicrystals: a primer. Second ed., Monographs on the Physics and Chemistry of Materials, Oxford University Press, Oxford, 1997.Google Scholar
[21] Janssen, T.. Aperiodic Schr¨odinger Operators. In: The mathematics of long-range aperiodic order (Waterloo, ON, 1995), NATO Adv. Sci. Ser. C Math. Phys. Sci., 489, Kluwer Academic Publishers, Dordrecht, 1997, pp. 269–306.Google Scholar
[22]Janssen, T. and Janner, A., Incommensurability in crystals. Adv. in Phys. 36(1987), no. 5, 519–624. http://dx.doi.org/10.1080/00018738700101052 Google Scholar
[23]Kelley, J. L., General topology. D. van Nostrand Company, Toronto-New York-London, 1955.Google Scholar
[24] Lagarias, J. C., Meyer's concept of quasicrystal and quasiregular sets. Commun. Math. Phys. 179(1996), no. 2, 365–376. http://dx.doi.org/10.1007/BF02102593 Google Scholar
[25] Lagarias, J. C., Geometric models for quasicrystals I. Delone sets of finite type. Discrete Comput. Geom. 21(1999), no. 2, 161–191. http://dx.doi.org/10.1007/PL00009413 Google Scholar
[26] Lagarias, J. C., Mathematical quasicrystals and the problem of diffraction. In: Directions in mathematical quasicrystals, CRM Monogr. Ser., 13, American Mathematical Society, Providence, RI, 2000, pp. 61–93.Google Scholar
[27] Lagarias, J. C. and Pleasants, P. A. B., Repetitive Delone sets and quasicrystals. Ergodic Theory Dynam. Systems 23(2003), no. 3, 831–867. http://dx.doi.org/10.1017/S0143385702001566 Google Scholar
[28] Lagarias, J. C., Local complexity of Delone sets and crystallinity. Canad. Math. Bull. 45(2002), no. 4, 634–652. http://dx.doi.org/10.4153/CMB-2002-058-0 Google Scholar
[29] Lenz, D., Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks. Commun. Math. Phys. 287(2009), no. 1, 225–258. http://dx.doi.org/10.1007/s00220-008-0594-2 Google Scholar
[30]Lenz, D. and Richard, C., Pure point diffraction and cut and project schemes for measures: the smooth case. Math. Z. 256(2007), no. 2, 347–378. http://dx.doi.org/10.1007/s00209-006-0077-0Google Scholar
[31] Lenz, D. and Stollmann, P., Delone dynamical systems and associated random operators. In: Operator algebras and mathematical physics (Constanta, 2001), Theta, Bucharest, pp. 267–285.Google Scholar
[32]Lenz, D. and Strungaru, N., Pure point spectrum for measure dyamical systems on locally compact Abelian groups. J. Math. Pures Appl. 92(2009), no. 4, 323–341 Google Scholar
[33] Meyer, Y., Algebraic numbers and harmonic analysis. North-Holland Mathematical Library, 2, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1972.Google Scholar
[34] Moody, R. V., ed., The mathematics of long-range aperiodic order. Proceedings of the NATO Advanced Study Institute held in Waterloo, ON, August 21–September 1, 1995. NATO Advanced Science Institutes Series C, 489, Kluwer Academic Publishers Group, Dordrecht, 1997.Google Scholar
[35] Moody, R. V., Meyer sets and their duals, in: The mathematics of long-range aperiodic order. Proceedings of the NATO Advanced Study Institute held inWaterloo, ON, August 21–September 1, 1995. NATO Advanced Science Institutes Series C, 489, Kluwer Academic Publishers Group, Dordrecht, 1997, pp. 403–441.Google Scholar
[36] Moody, R. V., Model sets: a survey. In: From quasicrystals to more complex systems. EDP Sciences, Les Ulis, and Springer, Berlin, 2000, pp. 145–166.Google Scholar
[37] Moody, R. V. and Strungaru, N., Point sets and dynamical systems in the autocorrelation topology. Canad. Math. Bull. 47(2004), no. 1, 82–99. http://dx.doi.org/10.4153/CMB-2004-010-8 Google Scholar
[38] Morse, M. and Hedlund, G. A., Symbolic dynamics. Amer. J. Math. 60(1938), no. 4, 815–866. http://dx.doi.org/10.2307/2371264 Google Scholar
[39] Morse, M., Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62(1940), 1–42. http://dx.doi.org/10.2307/2371431 Google Scholar
[40] Müller, P. and Richard, C., Ergodic properties of randomly coloured point sets. arxiv:1005.4884.Google Scholar
[41]Patera, J. (ed.), Quasicrystals and discrete geometry. Proceedings of the Fall Programme held at the University of Toronto, Toronto, ON, 1995. Fields Institute Monographs, 10, American Mathematical Society, Providence, RI, 1998.Google Scholar
[42]Pedersen, G.K., Analysis now. Graduate Texts in Mathematics, 118, Springer-Verlag, New York, 1989.Google Scholar
[43]Queffélec, M., Substitution dynamical systems—spectral analysis. Lecture Notes in Mathematics, 1294, Springer-Verlag, Berlin, 1987.Google Scholar
[44]Schlottmann, M., Generalized model sets and dynamical systems. In: Directions in mathematical quasicrystals, CRM Monogr. Ser., 13, American Mathematical Society, Providence, RI, pp. 143–159.Google Scholar
[45]Senechal, M., Quasicrystals and geometry. Cambridge University Press, Cambridge, 1995.Google Scholar
[46] Shechtman, D., Blech, I., Gratias, D., and Cahn, J.W., Metallic phase with long-range orientational order and no translational symmetry Phys. Rev. Lett. 53(1984), no. 20, 1951–1953.Google Scholar
[47] Solomyak, B., Dynamics of self-similar tilings. Ergodic Theory Dynam. Systems 17(1997), no. 3, 695–738; Erratum: Ibid. 19(1999), no. 6, 1685. http://dx.doi.org/10.1017/S0143385797084988 Google Scholar
[48] Solomyak, B., Spectrum of dynamical systems arising from Delone sets. In: Quasicrystals and discrete geometry (Toronto, ON, 1995), Fields Inst. Monogr., 10, American Mathematical Society, Providence, RI, 1998, pp. 265–275.Google Scholar
[49] Strungaru, N., Almost periodic measures and long-range order in Meyer sets. Discrete Comput. Geom. 33(2005), no. 3,483–505. http://dx.doi.org/10.1007/s00454-004-1156-9 Google Scholar
[50] Strungaru, N., On the spectrum of a Meyer set. arxiv:1003.3019.Google Scholar
[51] Zaidman, S., Almost periodic functions in abstract spaces. Research Notes in Mathematics, 126, Pitman, Boston, MA, 1985.Google Scholar