Journal of Statistical Physics

, Volume 174, Issue 3, pp 519–535 | Cite as

Diffraction of Return Time Measures

  • M. KesseböhmerEmail author
  • A. Mosbach
  • T. Samuel
  • M. Steffens


Letting T denote an ergodic transformation of the unit interval and letting \(f:[0,1)\rightarrow \mathbb {R}\) denote an observable, we construct the f-weighted return time measure \(\mu _y\) for a reference point \(y\in [0,1)\) as the weighted Dirac comb with support in \(\mathbb {Z}\) and weights \(f\circ T^z(y)\) at \(z\in \mathbb {Z},\) and if T is non-invertible, then we set the weights equal to zero for all \(z < 0.\) Given such a Dirac comb, we are interested in its diffraction spectrum and analyse it for the dependence on the underlying transformation. Under certain regularity conditions imposed on the interval map and the observable we explicitly calculate the diffraction of \(\mu _{y}\) which consists of a trivial atom and an absolutely continuous part, almost surely with respect to \(y.\) This contrasts what occurs in the setting of regular model sets arising from cut and project schemes and deterministic incommensurate structures. As a prominent example of non-mixing transformations, we consider rigid rotations. In this situation we observe that the diffraction of \(\mu _{y}\) is pure point, almost surely with respect to \(y\) and, if the rotation number is irrational and the observable is Riemann integrable, then the diffraction of \(\mu _{y}\) is independent of \(y.\) Finally, for a converging sequence of rotation numbers, we provide new results concerning the limiting behaviour of the associated diffractions.


Aperiodic order Autocorrelation Diffraction Transformations of the unit interval Rigid rotations 

Mathematics Subject Classification

43A25 52C23 37E05 37A25 37A45 



Part of this work was completed while the authors were visiting the Mittag-Leffler Institute as part of the Research Program Fractal Geometry and Dynamics. We are extremely grateful to the organisers and staff for their very kind hospitality, financial support and stimulating atmosphere. The authors also wish to thank M. Baake and N. Strungaru for many interesting and insightful discussions. Finally, the authors are grateful to MINTernational for providing financial support for research visits between Universität Bremen and California Polytechnic State University.


  1. 1.
    Argabright, L., Gil de Lamadrid, J.: Fourier Analysis of Unbounded Measures on Locally Compact Abelian Groups. Memoirs of the American Mathematical Society, No. 145. American Mathematical Society, Providence (1974)Google Scholar
  2. 2.
    Baake, M., Grimm, U.: Kinematic diffraction from a mathematical viewpoint. Z. Kristallogr. 226(9), 711–725 (2011)Google Scholar
  3. 3.
    Baake, M., Grimm, U.: Aperiodic Order. Cambridge University Press, CPI Group Ltd., London (2013)zbMATHGoogle Scholar
  4. 4.
    Baake, M., Höffe, M.: Diffraction of random tilings: some rigorous results. J. Stat. Phys. 99(1–2), 216–261 (2000)ADSMathSciNetzbMATHGoogle Scholar
  5. 5.
    Baake, M., Lenz, D.: Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra. Ergod. Theory Dyn. Syst. 24(6), 1867–1893 (2004)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Baake, M., Moody, R.V.: Weighted Dirac combs with pure point diffraction. J. Reine Angew. Math. 573, 61–94 (2004)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Beckus, S., Pogorzelski, F.: Delone dynamical systems and spectral convergence. Ergod. Theory Dyn. Syst.
  8. 8.
    Berend, D., Radin, C.: Are there chaotic tilings? Commun. Math. Phys. 152(2), 215–219 (1993)ADSMathSciNetzbMATHGoogle Scholar
  9. 9.
    Berg, C., Forst, G.: Potential Theory on Locally Compact Abelian Groups. Ergebnisse der Mathamatik und ihrer Grenzbebiete, Band. 87. Springer, Berlin (1975)Google Scholar
  10. 10.
    Boshernitzan, M.: A condition for minimal interval exchange maps to be uniquely ergodic. Duke Math. J. 52(3), 723–752 (1985)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Einsiedler, M., Ward, T.: Ergodic Theory with a View Towards Number Theory. Graduate Texts in Mathematics, vol. 259. Springer, London (2011)zbMATHGoogle Scholar
  12. 12.
    Gil de Lamadrid, J., Argabright, L.: Almost periodic measures. Mem. Am. Math. Soc. 85(428), vi+219 (1990)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gouéré, J.-B.: Diffraction and Palm measure of point processes (2002)Google Scholar
  14. 14.
    Hof, A.: On diffraction by aperiodic structures. Commun. Math. Phys. 169(1), 25–43 (1995)ADSMathSciNetzbMATHGoogle Scholar
  15. 15.
    Hofbauer, F., Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180(1), 119–140 (1982)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ishimasa, T., Nissen, H.-U., Fukano, Y.: New ordered state between crystalline and amorphous in Ni-Cr particles. Phys. Rev. Lett. 55, 511–513 (1985)ADSGoogle Scholar
  17. 17.
    Kellendonk, J., Lenz, D., Savinien, J. (eds.): Mathematics of Aperiodic Order. Progress in Mathematics, vol. 309. Birkhäuser/Springer, Basel (2015)zbMATHGoogle Scholar
  18. 18.
    Keller, G.: On the rate of convergence to equilibrium in one-dimensional systems. Commun. Math. Phys. 96(2), 181–193 (1984)ADSMathSciNetzbMATHGoogle Scholar
  19. 19.
    Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. A Wiley-Interscience publication. Wiley, New York (1974)Google Scholar
  20. 20.
    Lenz, D.: An autocorrelation and discrete spectrum for dynamical systems on metric spaces. arXiv:1608.05636 (2016)
  21. 21.
    Lenz, D., Strungaru, N.: Pure point spectrum for measure dynamical systems on locally compact Abelian groups. J. Math. Pures Appl. (9) 92(4), 323–341 (2009)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Lenz, D., Strungaru, N.: On weakly almost periodic measures. Trans. Am. Math. Soc.
  23. 23.
    Lindenstrauss, E.: Pointwise theorems for amenable groups. Invent. Math. 146(2), 259–295 (2001)ADSMathSciNetzbMATHGoogle Scholar
  24. 24.
    Moody, R.V. (ed.): The Mathematics of Long-Range Aperiodic Order. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 489. Kluwer Academic Publishers Group, Dordrecht (1997)Google Scholar
  25. 25.
    Moody, R.V.: Mathematical quasicrystals: a tale of two topologies. In: XIVth International Congress on Mathematical Physics, pp. 68–77. World Scientific Publishing, Hackensack (2005)Google Scholar
  26. 26.
    Mozes, S.: Tilings, substitution systems and dynamical systems generated by them. J. d’Anal. Math. 53(1), 139–186 (1989)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Müller, P., Richard, C.: Ergodic properties of randomly coloured point sets. Can. J. Math. 65(2), 349–402 (2013)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Richard, C.: Dense Dirac combs in Euclidean space with pure point diffraction. J. Math. Phys. 44(10), 4436–4449 (2003)ADSMathSciNetzbMATHGoogle Scholar
  29. 29.
    Richard, C., Strungaru, N.: Pure point diffraction and Poisson summation. Ann. Henri Poincaré 18(12), 3903–3931 (2017)ADSMathSciNetzbMATHGoogle Scholar
  30. 30.
    Richard, C., Strungaru, N.: A short guide to pure point diffraction in cut-and-project sets. J. Phys. A 50(15), 154003, 25 (2017)Google Scholar
  31. 31.
    Rudin, W.: Fourier Analysis on Groups. A Wiley-Interscience publication. Wiley, New York (1990)Google Scholar
  32. 32.
    Schlottmann, M.: Generalized model sets and dynamical systems. In: Directions in Mathematical Quasicrystals. CRM Monograph Series, vol. 13, pp. 143–159. American Mathematical Society, Providence (2000)Google Scholar
  33. 33.
    Senechal, M.: Quasicrystals and Geometry. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  34. 34.
    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)ADSGoogle Scholar
  35. 35.
    Solomyak, B.: Dynamics of self-similar tilings. Ergod. Theory Dyn. Syst. 17(3), 695–738 (1997)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics, vol. 79. Springer, New York (1982)Google Scholar

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Authors and Affiliations

  1. 1.FB 3 – Mathematik und InformatikUniversität BremenBremenGermany
  2. 2.Mathematics DepartmentCalifornia Polytechnic State UniversitySan Luis ObispoUSA
  3. 3.School of MathematicsUniversity of BirminghamBirminghamUK

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