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A Metric Characterisation of Repulsive Tilings

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Abstract

A tiling of \({\mathbb {R}}^d\) is repulsive if no r-patch can repeat arbitrarily close to itself, relative to r. This is a characteristic property of aperiodic order, for a non-repulsive tiling has arbitrarily large locally periodic patterns. We consider a non-periodic, repetitive tiling T of \({\mathbb {R}}^d\), with finite local complexity. From a spectral triple built on the discrete hull \(\varXi \) of T and its Connes distance, we derive two metrics \(d_{\mathrm{sup}}\) and \(d_{\mathrm{inf}}\) on \(\varXi \). We show that T is repulsive if and only if \(d_{\mathrm{sup}}\) and \(d_{\mathrm{inf}}\) are Lipschitz equivalent. This generalises the previous works on subshifts by J. Kellendonk, D. Lenz, and the author.

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Notes

  1. Sometimes also called a puncture, so that one talks about punctured tilings.

  2. See Remark 2.1.

  3. \(L_p\) is locally derived from T, but T might not be locally derived from \(L_p\), nor from L (think for instance of a Wang tiling if one chooses unspecified marker locations).

  4. Closed and open sets.

  5. \(d_{\mathrm{inf}}\) is an ultra-metric, and \(d_{\mathrm{sup}}\) is valued in \([0,+\infty ]\).

References

  1. Baake, M., Grimm, U.: Aperiodic Order. Volume 1. Encyclopedia of Mathematics and Its Applications, vol. 149. Cambridge University Press, Cambridge (2013)

    Google Scholar 

  2. Connes, A.: Noncommutative Geometry. Academic Press, San Diego (1994)

    MATH  Google Scholar 

  3. Christensen, E., Ivan, C.: Sums of two-dimensional spectral triples. Math. Scand. 100, 35–60 (2007)

    MATH  MathSciNet  Google Scholar 

  4. Christensen, E., Ivan, C., Lapidus, M.L.: Dirac operators and spectral triples for some fractal sets built on curves. Adv. Math. 217(1), 42–78 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Durand, F.: Linearly recurrent subshifts have a finite number of non-periodic subshift factors. Ergodic Theory Dyn. Syst. 20, 1061–1078 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Durand, F.: Corrigendum and addendum to [5]. Ergodic Theory Dyn. Syst. 23, 663–669 (2003)

    MathSciNet  Google Scholar 

  7. Frank, N., Sadun, L.: Fusion: a general framework for hierarchical tilings of \({\mathbb{R}}^d\). Geom. Dedicata 171, 149–186 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  8. Glen, A., Justin, J., Widmer, S., Zamboni, L.Q.: Palindromic richness. Eur. J. Comb. 30, 510–531 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Guido, D., Isola, T.: Dimensions and singular traces for spectral triples, with applications for fractals. J. Funct. Anal. 203, 362–400 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Guido, D., Isola, T.: Dimension and spectral triples for fractals in \({\mathbb{R}}^n\). In: Advances in Operator Algebras and Mathematical Physics. Theta Series in Advanced Mathematics, vol. 5, pp. 89–108. Theta, Bucarest (2005)

  11. Forsyth, M., Jayakumar, A., Shallit, J.: Remarks on privileged words. Acta Myol. 22(1), 1–4 (2003). arXiv:1306.6768

  12. Frid, A.E., Puzynina, S., Zamboni, L.: On palindromic factorization of words. Adv. Appl. Math. 50, 737–748 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kellendonk, J., Savinien, J.: Spectral triples and characterization of aperiodic order. Proc. Lond. Math. Soc. 104, 123–157 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kellendonk, J., Lenz, D., Savinien, J.: A characterisation of subshifts with bounded powers. Discrete Math. 313, 2881–2894 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lagarias, J., Pleasants, P.: Repetitive Delone sets and quasicrystals. Ergodic Theory Dyn. Syst. 23, 831–867 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lenz, D.: Aperiodic linearly repetitive Delone sets are densely repetitive. Discrete Comput. Geom. 31, 323–326 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Patera J. (ed.): Quasicrystals and Discrete Geometry, Toronto, ON, 1995. Fields Institute Monographs, vol. 10. American Mathematical Society, Providence, RI (1998)

  18. Pearson, J., Bellissard, J.: Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets. J. Noncommut. Geom. 3, 447–481 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. Peltomäki, J.: Introducing privileged words: privileged complexity of Sturmian words. Theor. Comput. Sci. 500, 57–67 (2013)

    Article  MATH  Google Scholar 

  20. Peltomäki, J.: Privileged factors in the Thue–Morse word—a comparison of privileged words and palindromes. Discrete Appl. Math. (2015). arXiv:1306.6768

  21. Rieffel, M.: Metrics on state spaces. Doc. Math. 4, 559–600 (1999)

    MATH  MathSciNet  Google Scholar 

  22. Rieffel, M.: Compact quantum metric spaces. In: Operator Algebras, Quantization, and Noncommutative Geometry. Contemporary Mathematics, vol. 365, pp. 315–330. American Mathematical Society, Providence, RI (2004)

  23. Solomyak, B.: Dynamics of self-similar tilings. Ergodic Theory Dyn. Syst. 17, 695–738 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  24. Solomyak, B.: Corrections to [23]. Ergodic Theory Dyn. Syst. 19, 1685 (1999)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The author would like to thank J. Kellendonk and D. Lenz for useful discussions and their encouragement to publish this work.

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Authors and Affiliations

  1. Université de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, 57045, Metz, France

    J. Savinien

  2. CNRS, Institut Elie Cartan de Lorraine, UMR 7502, 57045, Metz, France

    J. Savinien

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  1. J. Savinien
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Correspondence to J. Savinien.

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Savinien, J. A Metric Characterisation of Repulsive Tilings. Discrete Comput Geom 54, 705–716 (2015). https://doi.org/10.1007/s00454-015-9719-5

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  • DOI: https://doi.org/10.1007/s00454-015-9719-5

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