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Annales Henri Poincaré

, Volume 19, Issue 10, pp 3053–3088 | Cite as

Tiling Deformations, Cohomology, and Orbit Equivalence of Tiling Spaces

  • Antoine Julien
  • Lorenzo Sadun
Article
  • 21 Downloads

Abstract

We study homeomorphisms of minimal and uniquely ergodic tiling spaces with finite local complexity (FLC), of which suspensions of (minimal and uniquely ergodic) d-dimensional subshifts are an example, and orbit equivalence of tiling spaces with (possibly) infinite local complexity (ILC). In the FLC case, we construct a cohomological invariant of homeomorphisms and show that all homeomorphisms are a combination of tiling deformations, maps homotopic to the identity (known as quasi-translations), and local equivalences (MLD). In the ILC case, we construct a cohomological invariant in the so-called weak cohomology and show that all orbit equivalences are combinations of tiling deformations, quasi-translations, and topological conjugacies. These generalize results of Parry and Sullivan to higher dimensions. We also show that homeomorphisms (FLC) or orbit equivalences (ILC) are completely parametrized by the appropriate cohomological invariants. Finally, we show that, under suitable cohomological conditions, continuous maps between tiling spaces are homotopic to compositions of tiling deformations and either local derivations (FLC) or factor maps (ILC).

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Notes

Acknowledgements

Work of the second author is partially supported by NSF Grant DMS-1101326. We thank Johannes Kellendonk and Christian Skau for helpful discussions.

References

  1. 1.
    Aliste-Prieto, J., Coronel, D., Cortez, M.I., Durand, F., Petite, S.: Linearly repetitive delone sets. In: Kellendonk, J., Lenz, D., Savinien, J. (eds.) Mathematics of Aperiodic Order, Volume 309 of Progress in Mathematics, pp. 195–222. Springer, Basel (2015)CrossRefGoogle Scholar
  2. 2.
    Bellissard, J.V.: Delone sets and material science: a program. In: Kellendonk, J., Lenz, D., Savinien, J. (eds.) Mathematics of Aperiodic Order. Progress in Mathematics, vol. 309. Birkhäuser, Basel (2015). https://doi.org/10.1007/978-3-0348-0903-0_11
  3. 3.
    Bellissard, J.: Modeling Liquids and Bulk Metallic Glasses (Oral Communication). Mathematics of Novel Materials, Mittag-Leffler Institute, Djursholm (2015)Google Scholar
  4. 4.
    Bellissard, J., Benedetti, R., Gambaudo, J.-M.: Spaces of tilings, finite telescopic approximations and gap-labeling. Commun. Math. Phys. 261(1), 1–41 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Boulmezaoud, H., Kellendonk, J.: Comparing different versions of tiling cohomology. Topol. Appl. 157(14), 2225–2239 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Boyle, M., Handelman, D.: Orbit equivalence, flow equivalence and ordered cohomology. Israel J. Math. 95, 169–210 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Clark, A., Sadun, L.: When size matters: subshifts and their related tiling spaces. Ergod. Theory Dyn. Syst. 23, 1043–1057 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Clark, A., Sadun, L.: When shape matters: deformations of tiling spaces. Ergod. Theory Dyn. Syst. 26(1), 69–86 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cortez, M., Durand, F., Petite, S.: Linearly repetitive delone systems have a finite number of nonperiodic delone system factors. Proc. Am. Math. Soc. 138(3), 1033–1046 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fogg, N.P.: Substitutions in Dynamics, Arithmetics and Combinatorics. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  11. 11.
    Frank, N., Sadun, L.: Fusion tilings with infinite local complexity. Preprint (2012)Google Scholar
  12. 12.
    Giordano, T., Putnam, I.F., Skau, C.F.: Topological orbit equivalence and \(C^*\)-crossed products. J. Reine Angew. Math. 469, 51–111 (1995)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hunton, J.: Oral communication. Workshop on aperiodic order, Leicester (2015)Google Scholar
  14. 14.
    Julien, A.: Complexity as a homeomorphism invariant for tiling spaces. Ann. Inst. Fourier 67(2), 539–577 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kellendonk, J.: Pattern-equivariant functions and cohomology. J. Phys. A 36(21), 5765–5772 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kellendonk, J.: Pattern equivariant functions, deformations and equivalence of tiling spaces. Ergod. Theory Dyn. Syst. 28(4), 1153–1176 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kellendonk, J., Putnam, I.F.: The Ruelle–Sullivan map for actions of \(\mathbb{R}^n\). Math. Ann. 334(3), 693–711 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kellendonk, J., Sadun, L.: Meyer sets, topological eigenvalues, and Cantor fiber bundles. J. Lond. Math. Soc. (2) 89(1), 114–130 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kwapisz, J.: Topological friction in aperiodic minimal \(\mathbb{{R}}^m\)-actions. Fund. Math. 207(2), 175–178 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Moore, C.C., Schochet, C.L.: Global Analysis on Foliated Spaces. Volume 9 of Mathematical Sciences Research Institute Publications, 2nd edn. Cambridge University Press, New York (2006)Google Scholar
  21. 21.
    Morse, M., Hedlund, G.A.: Symbolic dynamics. Am. J. Math. 60(4), 815–866 (1938)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Parry, B., Sullivan, D.: A topological invariant of flows on \(1\)-dimensional spaces. Topology 14(4), 297–299 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Parry, W., Tuncel, S.: Classification Problems in Ergodic Theory, Volume 67 of London Mathematical Society Lecture Note Series, Statistics: Textbooks and Monographs, 41. Cambridge University Press, Cambridge (1982)Google Scholar
  24. 24.
    Parry, W., Tuncel, S.: On the stochastic and topological structure of Markov chains. Bull. Lond. Math. Soc. 14(1), 16–27 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Petersen, K.: Factor maps between tiling dynamical systems. Forum Math. 11, 503–512 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Radin, C.: The pinwheel tilings of the plane. Ann. Math. 26, 289–306 (1994)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Radin, C., Sadun, L.: Isomorphism of hierarchical structures. Ergod. Theory Dyn. Syst. 21(4), 1239–1248 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Rand, B., Sadun, L.: An approximation theorem for maps between tiling spaces. Disc. Cont. Dynam. Syst. 29, 323–326 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Sadun, L.: Topology of Tiling Spaces. Volume 46 of University Lecture Series. American Mathematical Society, Providence (2008)Google Scholar
  30. 30.
    Sadun, L.: Cohomology of hierarchical tilings. In: Kellendonk, J., Lenz, D., Savinien, J. (eds.) Mathematics of Aperiodic Order. Progress in Mathematics, vol. 309. Birkhäuser, Basel (2015).  https://doi.org/10.1007/978-3-0348-0903-0_3

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Nord UniversityLevangerNorway
  2. 2.Department of MathematicsUniversity of TexasAustinUSA

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