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).
Similar content being viewed by others
References
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)
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
Bellissard, J.: Modeling Liquids and Bulk Metallic Glasses (Oral Communication). Mathematics of Novel Materials, Mittag-Leffler Institute, Djursholm (2015)
Bellissard, J., Benedetti, R., Gambaudo, J.-M.: Spaces of tilings, finite telescopic approximations and gap-labeling. Commun. Math. Phys. 261(1), 1–41 (2006)
Boulmezaoud, H., Kellendonk, J.: Comparing different versions of tiling cohomology. Topol. Appl. 157(14), 2225–2239 (2010)
Boyle, M., Handelman, D.: Orbit equivalence, flow equivalence and ordered cohomology. Israel J. Math. 95, 169–210 (1996)
Clark, A., Sadun, L.: When size matters: subshifts and their related tiling spaces. Ergod. Theory Dyn. Syst. 23, 1043–1057 (2003)
Clark, A., Sadun, L.: When shape matters: deformations of tiling spaces. Ergod. Theory Dyn. Syst. 26(1), 69–86 (2006)
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)
Fogg, N.P.: Substitutions in Dynamics, Arithmetics and Combinatorics. Springer, Berlin (2002)
Frank, N., Sadun, L.: Fusion tilings with infinite local complexity. Preprint (2012)
Giordano, T., Putnam, I.F., Skau, C.F.: Topological orbit equivalence and \(C^*\)-crossed products. J. Reine Angew. Math. 469, 51–111 (1995)
Hunton, J.: Oral communication. Workshop on aperiodic order, Leicester (2015)
Julien, A.: Complexity as a homeomorphism invariant for tiling spaces. Ann. Inst. Fourier 67(2), 539–577 (2017)
Kellendonk, J.: Pattern-equivariant functions and cohomology. J. Phys. A 36(21), 5765–5772 (2003)
Kellendonk, J.: Pattern equivariant functions, deformations and equivalence of tiling spaces. Ergod. Theory Dyn. Syst. 28(4), 1153–1176 (2008)
Kellendonk, J., Putnam, I.F.: The Ruelle–Sullivan map for actions of \(\mathbb{R}^n\). Math. Ann. 334(3), 693–711 (2006)
Kellendonk, J., Sadun, L.: Meyer sets, topological eigenvalues, and Cantor fiber bundles. J. Lond. Math. Soc. (2) 89(1), 114–130 (2014)
Kwapisz, J.: Topological friction in aperiodic minimal \(\mathbb{{R}}^m\)-actions. Fund. Math. 207(2), 175–178 (2010)
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)
Morse, M., Hedlund, G.A.: Symbolic dynamics. Am. J. Math. 60(4), 815–866 (1938)
Parry, B., Sullivan, D.: A topological invariant of flows on \(1\)-dimensional spaces. Topology 14(4), 297–299 (1975)
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)
Parry, W., Tuncel, S.: On the stochastic and topological structure of Markov chains. Bull. Lond. Math. Soc. 14(1), 16–27 (1982)
Petersen, K.: Factor maps between tiling dynamical systems. Forum Math. 11, 503–512 (1999)
Radin, C.: The pinwheel tilings of the plane. Ann. Math. 26, 289–306 (1994)
Radin, C., Sadun, L.: Isomorphism of hierarchical structures. Ergod. Theory Dyn. Syst. 21(4), 1239–1248 (2001)
Rand, B., Sadun, L.: An approximation theorem for maps between tiling spaces. Disc. Cont. Dynam. Syst. 29, 323–326 (2011)
Sadun, L.: Topology of Tiling Spaces. Volume 46 of University Lecture Series. American Mathematical Society, Providence (2008)
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
Acknowledgements
Work of the second author is partially supported by NSF Grant DMS-1101326. We thank Johannes Kellendonk and Christian Skau for helpful discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Dmitry Dolgopyat.
Rights and permissions
About this article
Cite this article
Julien, A., Sadun, L. Tiling Deformations, Cohomology, and Orbit Equivalence of Tiling Spaces. Ann. Henri Poincaré 19, 3053–3088 (2018). https://doi.org/10.1007/s00023-018-0713-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-018-0713-3