Skip to main content Accessibility help
×
×
Home

Equicontinuous Delone Dynamical Systems

  • Johannes Kellendonk (a1) and Daniel Lenz (a2)
Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Equicontinuous Delone Dynamical Systems
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Equicontinuous Delone Dynamical Systems
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Equicontinuous Delone Dynamical Systems
      Available formats
      ×
Copyright
References
Hide All
[1] Auslander, J., Minimal flows and their extensions. North-Holland Mathematics Studies, 153, North-Holland Publishing Co., Amsterdam, 1988.
[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, 18671893. http://dx.doi.org/10.1017/S0143385704000318
[3] Baake, M., Deformation of Delone dynamical systems and pure point diffraction. J. Fourier Anal. Appl. 11(2005), no. 2, 125150. http://dx.doi.org/10.1007/s00041-005-4021-1
[4] Baake, M.,Lenz, D., and Moody, R. V., Characterization of model sets by dynamical systems. Ergodic Theory Dynam. Systems 27(2007), no. 2, 341382. http://dx.doi.org/10.1017/S0143385706000800
[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, 6177. http://dx.doi.org/10.1007/s11005-007-0186-7
[6] Baake, M. and Moody, R. V. (eds), Directions in mathematical quasicrystals. CRM Monograph Series, 13, American Mathematical Society, Providence, RI, 2000.
[7] Baake, M., Weighted Dirac combs with pure point diffraction. J. Reine Angew. Math. 573(2004), 6194. http://dx.doi.org/10.1515/crll.2004.064
[8]Barge, M. and Diamond, B., Proximality in Pisot tiling spaces. Fund. Math. 194(2007), no. 3, 191238. http://dx.doi.org/10.4064/fm194-3-1
[9] Barge, M. and Kellendonk, J., Proximality and pure point spectrum for tiling dynamical systems. arxiv:1108.4065.
[10] Barge, M. and Olimb, C., Asymptotic structure in substitution tiling spaces. arxiv:1101.4902.
[11]Barge, M. and Smith, M., Augmented dimension groups and ordered cohomology. Ergodic Theory Dynam. Systems 29(2009), no. 1, 135. http://dx.doi.org/10.1017/S0143385708080449
[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.
[13] Córdoba, A., La formule sommatoire de Poisson. C. R. Acad. Sci. Paris, Sér. I Math. 306(1988), no. 8, 373376.
[14]Forrest, A. H., Hunton, J. R., and Kellendonk, J., Topological invariants for projection method patterns. Mem. Amer. Math. Soc. 159(2002), no. 758.
[15] Gil de Lamadrid, J. and Argabright, L. N., Almost periodic measures. Mem. Amer. Math. Soc. 85(1990), no. 428.
[16] Gouéré, J.-B., Diffraction et mesure de Palm des processus ponctuels. C. R. Acad. Sci. 336(2003), no. 1, 5762.
[17] Gouéré, J.-B., Quasicrystals and almost periodicity. Commun. Math. Phys. 255(2005), no. 3, 655681. http://dx.doi.org/10.1007/s00220-004-1271-8
[18]Hof, A., On diffraction by aperiodic structures. Commun. Math. Phys. 169(1995), no. 1, 2543. http://dx.doi.org/10.1007/BF02101595
[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, 511513.
[20]Janot, C., Quasicrystals: a primer. Second ed., Monographs on the Physics and Chemistry of Materials, Oxford University Press, Oxford, 1997.
[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. 269306.
[22]Janssen, T. and Janner, A., Incommensurability in crystals. Adv. in Phys. 36(1987), no. 5, 519624. http://dx.doi.org/10.1080/00018738700101052
[23]Kelley, J. L., General topology. D. van Nostrand Company, Toronto-New York-London, 1955.
[24] Lagarias, J. C., Meyer's concept of quasicrystal and quasiregular sets. Commun. Math. Phys. 179(1996), no. 2, 365376. http://dx.doi.org/10.1007/BF02102593
[25] Lagarias, J. C., Geometric models for quasicrystals I. Delone sets of finite type. Discrete Comput. Geom. 21(1999), no. 2, 161191. http://dx.doi.org/10.1007/PL00009413
[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. 6193.
[27] Lagarias, J. C. and Pleasants, P. A. B., Repetitive Delone sets and quasicrystals. Ergodic Theory Dynam. Systems 23(2003), no. 3, 831867. http://dx.doi.org/10.1017/S0143385702001566
[28] Lagarias, J. C., Local complexity of Delone sets and crystallinity. Canad. Math. Bull. 45(2002), no. 4, 634652. http://dx.doi.org/10.4153/CMB-2002-058-0
[29] Lenz, D., Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks. Commun. Math. Phys. 287(2009), no. 1, 225258. http://dx.doi.org/10.1007/s00220-008-0594-2
[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, 347378. http://dx.doi.org/10.1007/s00209-006-0077-0
[31] Lenz, D. and Stollmann, P., Delone dynamical systems and associated random operators. In: Operator algebras and mathematical physics (Constanta, 2001), Theta, Bucharest, pp. 267285.
[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, 323341
[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.
[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.
[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. 403441.
[36] Moody, R. V., Model sets: a survey. In: From quasicrystals to more complex systems. EDP Sciences, Les Ulis, and Springer, Berlin, 2000, pp. 145166.
[37] Moody, R. V. and Strungaru, N., Point sets and dynamical systems in the autocorrelation topology. Canad. Math. Bull. 47(2004), no. 1, 8299. http://dx.doi.org/10.4153/CMB-2004-010-8
[38] Morse, M. and Hedlund, G. A., Symbolic dynamics. Amer. J. Math. 60(1938), no. 4, 815866. http://dx.doi.org/10.2307/2371264
[39] Morse, M., Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62(1940), 142. http://dx.doi.org/10.2307/2371431
[40] Müller, P. and Richard, C., Ergodic properties of randomly coloured point sets. arxiv:1005.4884.
[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.
[42]Pedersen, G.K., Analysis now. Graduate Texts in Mathematics, 118, Springer-Verlag, New York, 1989.
[43]Queffélec, M., Substitution dynamical systems—spectral analysis. Lecture Notes in Mathematics, 1294, Springer-Verlag, Berlin, 1987.
[44]Schlottmann, M., Generalized model sets and dynamical systems. In: Directions in mathematical quasicrystals, CRM Monogr. Ser., 13, American Mathematical Society, Providence, RI, pp. 143159.
[45]Senechal, M., Quasicrystals and geometry. Cambridge University Press, Cambridge, 1995.
[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, 19511953.
[47] Solomyak, B., Dynamics of self-similar tilings. Ergodic Theory Dynam. Systems 17(1997), no. 3, 695738; Erratum: Ibid. 19(1999), no. 6, 1685. http://dx.doi.org/10.1017/S0143385797084988
[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. 265275.
[49] Strungaru, N., Almost periodic measures and long-range order in Meyer sets. Discrete Comput. Geom. 33(2005), no. 3, 483505. http://dx.doi.org/10.1007/s00454-004-1156-9
[50] Strungaru, N., On the spectrum of a Meyer set. arxiv:1003.3019.
[51] Zaidman, S., Almost periodic functions in abstract spaces. Research Notes in Mathematics, 126, Pitman, Boston, MA, 1985.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed