Advertisement

Geometriae Dedicata

, Volume 171, Issue 1, pp 149–186 | Cite as

Fusion: a general framework for hierarchical tilings of \(\mathbb{R }^d\)

  • Natalie Priebe Frank
  • Lorenzo Sadun
Original Paper

Abstract

We introduce a formalism for handling general spaces of hierarchical tilings, a category that includes substitution tilings, Bratteli–Vershik systems, S-adic transformations, and multi-dimensional cut-and-stack transformations. We explore ergodic, spectral and topological properties of these spaces. We show that familiar properties of substitution tilings carry over under appropriate assumptions, and give counter-examples where these assumptions are not met. For instance, we exhibit a minimal tiling space that is not uniquely ergodic, with one ergodic measure having pure point spectrum and another ergodic measure having mixed spectrum. We also exhibit a 2-dimensional tiling space that has pure point measure-theoretic spectrum but is topologically weakly mixing.

Keywords

Self-similar Substitution Mixing Dynamical spectrum Invariant measures 

Mathematics Subject Classification (1991)

Primary: 37B50 Secondary: 52C23 37A25 37B10 

Notes

Acknowledgments

We thank Mike Boyle, Lewis Bowen, Kariane Calta, Amos Nevo, E. Arthur Robinson, Jr. and Boris Solomyak for helpful discussions. The work of L.S. is partially supported by NSF Grants DMS-0701055 and DMS-1101326.

References

  1. 1.
    Adams, T.: Smorodinsky’s conjecture on rank-one mixing. Proc. Am. Math. Soc. 126(3), 739–744 (1998)CrossRefMATHGoogle Scholar
  2. 2.
    Adams, T., Silva, C.: \({\mathbb{Z}}^d\) staircase actions. Ergod. Theory Dyn. Syst. 19(4), 837–850 (1999)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Aliste-Prieto, J., Coronel, D.: Tower systems for Linearly repetitive Delone sets. Ergod. Theory Dyn. Syst. 31, 1595–1618 (2011)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Anderson, J., Putnam, I.F.: Topological invariants for substitution tilings and their C*-algebras. Ergod. Theory Dyn. Syst. 18, 509–537 (1998)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Arnoux, P., Ornstein, D.S., Weiss, B.: Cutting and stacking, interval exchanges and geometric models. Israel J. Math. 50(1–2), 160–168 (1985)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Barge, M., Diamond, B.: Cohomology in one-dimensional substitution tiling spaces. Proc. Am. Math. Soc. 136(6), 2183–2191 (2008)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Barge, M., Diamond, B., Hunton, J., Sadun, L.: Cohomology of substitution tiling spaces. Ergod. Theory Dyn. Syst. 30, 1607–1627 (2010)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Bressaud, X., Durand, F., Maass, A.: On the eigenvalues of finite rank Bratteli–Vershik dynamical systems. J. Lond. Math. Soc. 2nd Ser. 72, 799–816 (2005)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Bellissard, J., Benedetti, R., Gambaudo, J.-M.: Spaces of tilings, finite telescopic approximations and gap-labeling. Commun. Math. Phys. 261(1), 1–41 (2006)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Bellissard, J., Julien, A., Savinien, J.: Tiling groupoids and Bratteli diagrams. Ann. Henri Poincaré 11(1), 69–99 (2010)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Berger, R.: The undecidability of the domino problem. Memoirs Am. Math. Soc. 66, 1–72 (1966)Google Scholar
  12. 12.
    Bezuglyi, S., Kwiatkowski, J., Medynets, K., Solomyak, B.: Finite rank Bratteli diagrams: structure of invariant measures. Trans. Am. Math. Soc. 365, 2637–2679 (2013)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Cortez, M., Gambaudo, J.-M., Maass, A.: Rotation topological factors of minimal \({\mathbb{Z}}^d\) actions on the Cantor set. Trans. Am. Math. Soc. 359, 2305–2315 (2007)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Chacon, R.V.: A geometric construction of measure preserving transformations. In: Fifth Berkeley Symposium Mathematical Statistics and Probability (Berkeley, CA 1965/55), vol 2: Contributions to Probability Theory, Part 2, pp. 335–360. University of California Press, Berkeley, CA (1967)Google Scholar
  15. 15.
    Clark, A., Sadun, L.: When size matters: subshifts and their related tiling spaces. Ergod. Theory Dyn. Syst. 23, 1043–1057 (2003)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Clark, A., Sadun, L.: When shape matters: deformations of tiling spaces. Ergod. Theory Dyn. Syst. 26, 69–86 (2006)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Cohen, R.: A Chacon \({\mathbb{R}}^2\)-action and proof of two-fold self-joining. Dissertation, Department of Mathematics, Bryn Mawr College (1993)Google Scholar
  18. 18.
    Danilenko, A.I., Silva, C.E.: Mixing rank-one actions of locally compact abelian groups. Ann. Inst. H. Poincaré Probab. Stat. 43(4), 375–398 (2007)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Dekking, F.M.: The spectrum of dynamical systems arising from substitutions of constant length. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41, 221–239 (1977)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Dekking, F.M., Keane, M.: Mixing properties of substitutions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 42, 23–33 (1978)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Downarowicz, T.: The Choquet simplex of invariant measures for minimal flows. Israel J. Math. 74, 241–256 (1991)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Durand, F.: Linearly recurrent subshifts have a finite number of non-periodic subshift factors. Ergod. Theory Dyn. Syst. 20, 1061–1078 (2000)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Ferenczi, S., Fisher, A.M., Talet, M.: Minimality and unique ergodicity of adic transformations. J. Anal. Math. 109, 1–31 (2009)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Fisher, A.M.: Nonstationary mixing and the unique ergodicity of adic transformations. Stoch. Dyn. 9, 335–391 (2009)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Frank, N.P.: A primer on substitutions tilings of Euclidean space. Expo. Math. 26(4), 295–326 (2008)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Frank, N.P., Robinson Jr, E.A.: Generalized \(\beta \)-expansions, substitution tilings, and local finiteness. Trans. Am. Math. Soc. 360(3), 1163–1177 (2008)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Frank, N.P., Solomyak, B.: A characterization of planar pseudo-self-similar tilings. Discrete Comput. Geom. 26(3), 289–306 (2001)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Frank, N.P., Sadun, L.: Fusion tilings with infinite local complexity. In: Topology Proceedings. arXiv:1201.3911 (2013)Google Scholar
  29. 29.
    Gähler, F., Maloney, G.: Cohomology of one-dimensional mixed substitution tiling spaces. Preprint 2011, arXiv:1112.1475 (2011)Google Scholar
  30. 30.
    Gardner, M.: Extraordinary nonperiodic tiling that enriches the theory of tiles. Sci. Am. 231, 116–119 (1977)CrossRefGoogle Scholar
  31. 31.
    Goodman-Strauss, C.: Matching rules and substitution tilings. Ann. Math. 147, 181–223 (1998)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Holton, C., Radin, C., Sadun, L.: Conjugacies for tiling dynamical systems. Commun. Math. Phys. 254(2), 343–359 (2005)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Host, B.: Valeurs propres de systèmes dynamiques definis par de substitutions de longueur variable. Ergod. Theory Dyn. Syst. 6, 529–540 (1986)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Jewett, R.I.: The prevalence of uniquely ergodic systems. J. Math. Mech. 19, 717–729 (1969/1970)Google Scholar
  35. 35.
    Johnson, A.S.A., Sahin, A.: Rank one and loosely Bernoulli actions in \({\mathbb{Z}}^d\). Ergod. Theory Dyn. Syst. 18(5), 1159–1172 (1998)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Kellendonk, J., Putnam, I.: Tilings, \(C^*\)-algebras, and \(K\)-theory. In: Directions in Mathematical Quasicrystals. CRM Monogr. Ser. 13, Amer. Math. Soc., pp. 177–206. Providence, RI (2000)Google Scholar
  37. 37.
    Kellendonk, J., Sadun, L.: Meyer sets, topological eigenvalues and Cantor fiber bundles. Preprint arXiv:1211.2250 to appear in Journal of London Mathematical Society (2013)Google Scholar
  38. 38.
    Krieger, W.: On unique ergodicity. In: Proceedings of the Sixth Berkeley Symposium Mathematical Statistics and Probability (Berkeley, California, 1970/1971), vol. II: Probability Theory, pp. 327–346. University of California Press, Bekeley, CA (1972)Google Scholar
  39. 39.
    Morse, M.: Recurrent geodesics on a surface of negative curvature. Trans. Am. Math. Soc. 22, 84–100 (1921)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Mossé, B.: Puissances de mots et reconnaissabilit des points fixes d’une substitution. Theor. Comput. Sci. 99(2), 327–334 (1992)CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Mozes, S.: Tilings, substitution systems and dynamical systems generated by them. J. Anal. Math. 53, 139–186 (1989)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Nevo, A.: Pointwise ergodic theorems for actions of groups. In: Handbook of Dynamical Systems, vol. 1B, pp. 871–982. Elsevier B. V, Amsterdam (2006)Google Scholar
  43. 43.
    Ormes, N., Radin, C., Sadun, L.: A homeomorphism invariant for substitution tiling spaces. Geom. Dedicata 90, 153–182 (2002)CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Ornstein, D.S., Weiss, B.: Ergodic theory of amenable group actions. I: the Rohlin lemma. Bull. Am. Math. Soc. 2(1), 161–164 (1980)CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    Petersen, K.: Ergodic Theory, Cambridge Studies in Advanced Mathematics 2. Cambridge University Press, Cambridge (1989)Google Scholar
  46. 46.
    Prouhet, E.: Mémoir sur quelques relations entre les puissances des nombres. C. R. Acad. Sci. Paris Sr. 1(33), 225 (1851)Google Scholar
  47. 47.
    Queffelec, M.: Substitution Dynamical Systems—Spectral Analysis. Springer, Berlin (1987)MATHGoogle Scholar
  48. 48.
    Radin, C.: The pinwheel tilings of the plane. Ann. Math. 139(3), 661–702 (1994)CrossRefMATHMathSciNetGoogle Scholar
  49. 49.
    Rand, B.: Pattern-equivariant cohomology of tiling spaces with rotations Ph.D. dissertation, University of Texas (2006)Google Scholar
  50. 50.
    Robinson, E.A.: Symbolic dynamics and tilings of \({\mathbb{R}}^d\). Proc. Sympos. Appl. Math. 20, 81–119 (2004)CrossRefGoogle Scholar
  51. 51.
    Robinson Jr, E.A., Sahin, A.A.: Rank-one \({\mathbb{Z}}^d\) actions and directional entropy. Ergod. Theory Dyn. Syst. 31, 285–299 (2011)CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    Robinson, R.: Undecidability and nonperiodicity for tilings of the plane. Invent. Math. 12, 177–209 (1971)CrossRefMATHMathSciNetGoogle Scholar
  53. 53.
    Rudolph, D.J.: The second centralizer of a Bernoulli shift is just its powers. Israel J. Math. 29(2–3), 167–178 (1978)CrossRefMATHMathSciNetGoogle Scholar
  54. 54.
    Sadun, L.: Tiling spaces are inverse limits. J. Math. Phys. 44(11), 5410–5414 (2003)CrossRefMATHMathSciNetGoogle Scholar
  55. 55.
    Sadun, L.: Topology of tiling spaces. University Lecture Series 46. American Mathematical Society (2008)Google Scholar
  56. 56.
    Sadun, L.: Exact regularity and the cohomology of tiling spaces. Ergod. Theory Dyn. Syst. 31, 1819–1834 (2011)CrossRefMATHMathSciNetGoogle Scholar
  57. 57.
    Solomyak, B.: Dynamics of self-similar tilings. Ergod. Theory Dyn. Syst. 17, 695–738, (1997). Errata. Ergod. Theory Dyn. Syst. 19, 1685 (1999)Google Scholar
  58. 58.
    Solomyak, B.: Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20(2), 265–279 (1998)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of MathematicsVassar CollegePoughkeepsieUSA
  2. 2.Department of MathematicsThe University of Texas at AustinAustinUSA

Personalised recommendations