Communications in Mathematical Physics

, Volume 357, Issue 3, pp 1071–1112 | Cite as

Self Affine Delone Sets and Deviation Phenomena

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Abstract

We study the growth of norms of ergodic integrals for the translation action on spaces coming from expansive, self-affine Delone sets. The linear map giving the self-affinity induces a renormalization map on the pattern space and we show that the rate of growth of ergodic integrals is controlled by the induced action of the renormalizing map on the cohomology of the pattern space up to boundary errors. We explore the consequences for the diffraction of such Delone sets, and explore in detail what the picture is for substitution tilings as well as for cut and project sets which are self-affine. We also explicitly compute some examples.

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References

  1. Ada04.
    Adamczewski, B.: Symbolic discrepancy and self-similar dynamics. Ann. Inst. Fourier (Grenoble) 54(7), 2201–2234 (2004)Google Scholar
  2. AP98.
    Anderson, J.E., Putnam, I.F.: Topological invariants for substitution tilings and their associated C *-algebras. Ergod. Theory Dyn. Syst. 18(3), 509–537 (1998)Google Scholar
  3. APCG11.
    Aliste-Prieto J., Coronel D., Gambaudo J.-M.: Rapid convergence to frequency for substitution tilings of the plane. Commun. Math. Phys. 306(2), 365–380 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. APCG13.
    Aliste-Prieto J., Coronel D., Gambaudo J.-M.: Linearly repetitive Delone sets are rectifiable. Ann. Inst. Henri Poincaré Anal. Non Linéaire 30(2), 275–290 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. BD08.
    Barge M., Diamond B.: Cohomology in one-dimensional substitution tiling spaces. Proc. Am. Math. Soc. 136(6), 2183–2191 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. BG13.
    Baake, M., Grimm, U.: Aperiodic order, vol. 1. In: Encyclopedia of Mathematics and its Applications, vol. 149. Cambridge University Press, Cambridge (2013) A mathematical invitation, With a foreword by Roger PenroseGoogle Scholar
  7. BK13a.
    Barge M., Kellendonk J.: Proximality and pure point spectrum for tiling dynamical systems. Mich. Math. J. 62(4), 793–822 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. BK13b.
    Barge M., Kellendonk J.: Proximality and pure point spectrum for tiling dynamical systems. Mich. Math. J. 62(4), 793–822 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. BKS12.
    Barge M., Kellendonk J., Schmieding S.: Maximal equicontinuous factors and cohomology for tiling spaces. Fund. Math. 218(3), 243–268 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. BS13.
    Bufetov A.I., Solomyak B.: Limit theorems for self-similar tilings. Commun. Math. Phys. 319(3), 761–789 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. Buf14.
    Bufetov A.I.: Finitely-additive measures on the asymptotic foliations of a Markov compactum. Mosc. Math. J. 14(2), 205–224, 426 (2014)MathSciNetMATHGoogle Scholar
  12. CF15.
    Cosentino, S., Flaminio, L.: Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds. ArXiv e-prints (2015)Google Scholar
  13. DF15.
    Dolgopyat, D., Fayad, B.: Limit theorems for toral translations. In: Hyperbolic Dynamics, Fluctuations and Large Deviations, Proceedings of Symposia in Pure Mathematics, vol. 89. American Mathematical Society, Providence, RI, pp. 227–277 (2015)Google Scholar
  14. DHL14.
    Delecroix V., Hubert P., Lelièvre S.: Diffusion for the periodic wind-tree model. Ann. Sci. Éc. Norm. Supér. (4) 47(6), 1085–1110 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. Dwo93.
    Dworkin S.: Spectral theory and x-ray diffraction. J. Math. Phys. 34(7), 2965–2967 (1993)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. FHK02.
    Forrest A., Hunton J., Kellendonk J.: Topological invariants for projection method patterns. Mem. Am. Math. Soc. 159(758), x+120 (2002)MathSciNetMATHGoogle Scholar
  17. FM13.
    Forni, G., Matheus, C.: Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. ArXiv e-prints (2013)Google Scholar
  18. For02.
    Forni G.: Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Ann. Math. (2) 155(1), 1–103 (2002)MathSciNetCrossRefMATHGoogle Scholar
  19. FSU15.
    Fra̧czek, K., Shi, R., Ulcigrai, C.: Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions. ArXiv e-prints (2015)Google Scholar
  20. GHK13.
    Gähler F., Hunton J., Kellendonk J.: Integral cohomology of rational projection method patterns. Algebr. Geom. Topol. 13(3), 1661–1708 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. Hat02.
    Hatcher A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)MATHGoogle Scholar
  22. HKW14.
    Haynes A., Kelly M., Weiss B.: Equivalence relations on separated nets arising from linear toral flows. Proc. Lond. Math. Soc. (3) 109(5), 1203–1228 (2014)MathSciNetCrossRefMATHGoogle Scholar
  23. HL04.
    Harriss E.O., Lamb J.S.W.: Canonical substitutions tilings of Ammann–Beenker type. Theor. Comput. Sci. 319(1–3), 241–279 (2004)MathSciNetCrossRefMATHGoogle Scholar
  24. Kal05.
    Kalugin P.: Cohomology of quasiperiodic patterns and matching rules. J. Phys. A 38(14), 3115–3132 (2005)MathSciNetCrossRefMATHGoogle Scholar
  25. Kel08.
    Kellendonk J.: Pattern equivariant functions, deformations and equivalence of tiling spaces. Ergod. Theory Dyn. Syst. 28(4), 1153–1176 (2008)MathSciNetCrossRefMATHGoogle Scholar
  26. Kon97.
    Kontsevich, M.: Lyapunov exponents and Hodge theory. The mathematical beauty of physics (Saclay, 1996). Adv. Ser. Math. Phys., vol. 24, pp. 318–332. World Scientific Publishing, River Edge, NJ (1997)Google Scholar
  27. KP06.
    Kellendonk J., Putnam I.F.: The Ruelle–Sullivan map for actions of \({{\mathbb{R}}^n}\). Math. Ann. 334(3), 693–711 (2006)MathSciNetCrossRefMATHGoogle Scholar
  28. Kwa11.
    Kwapisz J.: Rigidity and mapping class group for abstract tiling spaces. Ergod. Theory Dyn. Syst. 31(6), 1745–1783 (2011)MathSciNetCrossRefMATHGoogle Scholar
  29. Len09.
    Lenz D.: Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks. Commun. Math. Phys. 287(1), 225–258 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  30. LMS02.
    Lee J.-Y., Moody R.V., Solomyak B.: Pure point dynamical and diffraction spectra. Ann. Henri Poincaré 3(5), 1003–1018 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. LW03.
    Lagarias J.C., Wang Y.: Substitution Delone sets. Discrete Comput. Geom. 29(2), 175–209 (2003)MathSciNetCrossRefMATHGoogle Scholar
  32. Mat95.
    Mattila, P.: Geometry of sets and measures in Euclidean spaces. Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge, Fractals and rectifiability (1995)Google Scholar
  33. MS06.
    Moore, C.C., Schochet, C.L.: Global analysis on foliated spaces. second ed., Mathematical Sciences Research Institute Publications, vol. 9. Cambridge University Press, New York (2006)Google Scholar
  34. Put15.
    Putnam, I.F.: Lecture Notes on Smale Spaces. (2015). http://www.math.uvic.ca/faculty/putnam/ln/Smale_spaces.pdf
  35. Rob07.
    Robinson, E.A., Jr.: A Halmos–von Neumann theorem for model sets, and almost automorphic dynamical systems. In: Dynamics, Ergodic Theory, and Geometry. Mathematical Sciences Research Institute Publications, vol. 54, pp. 243–272. Cambridge University Press, Cambridge (2007)Google Scholar
  36. Sad03.
    Sadun L.: Tiling spaces are inverse limits. J. Math. Phys. 44(11), 5410–5414 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  37. Sad07.
    Sadun L.: Pattern-equivariant cohomology with integer coefficients. Ergod. Theory Dyn. Syst. 27(6), 1991–1998 (2007)MathSciNetCrossRefMATHGoogle Scholar
  38. Sad08.
    Sadun, L.: Topology of tiling spaces. University Lecture Series, vol. 46. American Mathematical Society, Providence, RI (2008)Google Scholar
  39. Sad11.
    Sadun L.: Exact regularity and the cohomology of tiling spaces. Ergod. Theory Dyn. Syst. 31(6), 1819–1834 (2011)MathSciNetCrossRefMATHGoogle Scholar
  40. SBGC84.
    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)ADSCrossRefGoogle Scholar
  41. Sch57.
    Schwartzman Sol: Asymptotic cycles. Ann. Math. (2) 66, 270–284 (1957)MathSciNetCrossRefMATHGoogle Scholar
  42. Sch03.
    Schwartzman S.: Higher dimensional asymptotic cycles. Can. J. Math. 55(3), 636–648 (2003)MathSciNetCrossRefMATHGoogle Scholar
  43. Sol97.
    Solomyak Boris: Dynamics of self-similar tilings. Ergod. Theory Dyn. Syst. 17(3), 695–738 (1997)MathSciNetCrossRefMATHGoogle Scholar
  44. Sol11.
    Solomon Y.: Substitution tilings and separated nets with similarities to the integer lattice. Israel J. Math. 181, 445–460 (2011)MathSciNetCrossRefMATHGoogle Scholar
  45. Sol14.
    Solomon Y.: A simple condition for bounded displacement. J. Math. Anal. Appl. 414(1), 134–148 (2014)MathSciNetCrossRefMATHGoogle Scholar
  46. ST16.
    Schmieding, S., Treviño, R.: Traces of random operators associated with self-affine Delone sets and Shubin’s formula. ArXiv e-prints (2016)Google Scholar
  47. Zor99.
    Zorich, A.: How do the leaves of a closed 1-form wind around a surface? Pseudoperiodic topology. American Mathematical Society Translations: Series 2, vol. 197, pp. 135–178. American Mathematical Society, Providence, RI (1999)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Northwestern UniversityEvanstonUSA
  2. 2.University of MarylandCollege ParkUSA

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