Skip to main content
SpringerLink
Log in

Diffraction of Random Noble Means Words

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

In this paper, several aspects of the random noble means substitution are studied. Beyond important dynamical facets such as the frequency of subwords and the computation of the topological entropy, the important issue of ergodicity is addressed. From the geometrical point of view, we outline a suitable cut and project setting for associated point sets and present results for the spectral analysis of the diffraction measure.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Baake, M., Grimm, U.: Aperiodic Order. A Mathematical Invitation, vol. 1. Cambridge University Press, Cambridge (2013)

  2. Berg, C., Forst, G.: Potential Theory on Locally Compact Abelian Groups. Springer, Berlin (1975)

    Book  MATH  Google Scholar 

  3. Cornfeld, I.P., Fomin, S.V., Sinai, Y.G.: Ergodic Theory. Springer, New York (1982)

    Book  MATH  Google Scholar 

  4. Etemadi, N.: An elementary proof of the strong law of large numbers. Z. Wahrscheinlichkeitsth. Verw. Geb. 55, 119–122 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fogg, N.P.: Substitutions in Dynamics, Arithmetics and Combinatorics. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  6. Godrèche, C., Luck, J.M.: Quasiperiodicity and randomness in tilings of the plane. J. Stat. Phys. 55, 1–28 (1989)

    Article  ADS  MATH  Google Scholar 

  7. Hof, A.: On diffraction by aperiodic structures. Commun. Math. Phys. 169, 25–43 (1995)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Kitchens, B.: Symbolic Dynamics: One-Sided, Two-Sided and Countable State Markov Shifts. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  9. Lang, S.: Real and Functional Analysis, 3rd edn. Springer, New York (1993)

    Book  MATH  Google Scholar 

  10. Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  11. Moll, M.: On a family of random noble means substitutions. Ph.D. thesis. http://pub.uni-bielefeld.de/publication/2637807 (2013). Accessed 25 Nov 2013

  12. Moody, R.V.: Meyer sets and their duals. In: Moody, R.V. (ed.) The Mathematics of Long-Range Aperiodic Order, NATO ASI Series C 489, pp. 403–449. Kluwer, Dordrecht (1997)

    Chapter  Google Scholar 

  13. Müller, P., Richard, C.: Ergodic properties of randomly coloured point sets. Can. J. Math. 65, 349–402 (2013). arXiv:1005.4884

  14. Nilsson, J.: On the entropy of a family of random substitutions. Mon. Math. 166, 1–15 (2012). arXiv:1103.4777

  15. Nilsson, J.: On the entropy of a two step random Fibonacci substitution. Entropy 15, 3312–3324 (2013). arXiv:1303.2526

  16. Parthasarathy, K.R.: Introduction to Probability and Measure, TRM. Hindustan Book Agency, New Delhi (2005)

    Google Scholar 

  17. Queffélec, M.: Substitution Dynamical Systems: Spectral Analysis, LNM 1294, 2nd edn. Springer, Berlin (2010)

    Book  Google Scholar 

  18. Schlottmann, M.: Cut-and-project sets in locally compact Abelian groups. In: Patera, J. (ed.) Quasicrystals and Discrete Geometry, Fields Institute Monographs, vol. 10, pp. 247–264. AMS, Providence (1998)

    Google Scholar 

  19. Seneta, E.: Non-negative Matrices and Markov Chains, 2nd rev. edn. Springer, New York (2006)

Download references

Acknowledgments

The author wishes to thank Michael Baake, Tobias Jakobi and Johan Nilsson for helpful discussions. This work is supported by the German Research Foundation (DFG) via the Collaborative Research Centre (CRC 701) through the faculty of Mathematics of Bielefeld University.

Author information

Authors and Affiliations

  1. Faculty of Mathematics, University of Bielefeld, Universitätsstr. 25, 33615 , Bielefeld, Germany

    Markus Moll

Authors
  1. Markus Moll
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Markus Moll.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Moll, M. Diffraction of Random Noble Means Words. J Stat Phys 156, 1221–1236 (2014). https://doi.org/10.1007/s10955-014-1047-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-014-1047-2

Keywords

Search

Navigation