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Rigidity and mapping class group for abstract tiling spaces

Published online by Cambridge University Press:  14 March 2011

JAROSLAW KWAPISZ
JAROSLAW KWAPISZ*
Affiliation:
Department of Mathematical Sciences, Montana State University, Bozeman MT 59717-2400, USA (email: jarek@math.montana.edu)
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Abstract

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We study abstract self-affine tiling actions, which are an intrinsically defined class of minimal expansive actions of ℝd on a compact space. They include the translation actions on the compact spaces associated to aperiodic repetitive tilings or Delone sets in ℝd. In the self-similar case, we show that the existence of a homeomorphism between tiling spaces implies conjugacy of the actions up to a linear rescaling. We also introduce the general linear group of an (abstract) tiling, prove its discreteness, and show that it is naturally isomorphic with the (pointed) mapping class group of the tiling space. To illustrate our theory, we compute the mapping class group for a five-fold symmetric Penrose tiling.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 6 , December 2011 , pp. 1745 - 1783
Copyright
Copyright © Cambridge University Press 2011

References

[AP98]Anderson, J. E. and Putnam, I. F.. Topological invariants for substitution tilings and their associated C*-algebras. Ergod. Th. & Dynam. Sys. 18(3) (1998), 509–537.CrossRefGoogle Scholar
[BD07]Barge, M. and Diamond, B.. Proximality in Pisot tiling spaces. Fund. Math. 194(3) (2007), 191–238.CrossRefGoogle Scholar
[BG03]Benedetti, R. and Gambaudo, J.-M.. On the dynamics of 𝒢-solenoids. Applications to Delone sets. Ergod. Th. & Dynam. Sys. 23 (2003), 673–691.CrossRefGoogle Scholar
[BK06]Barge, M. and Kwapisz, J.. Geometric theory of unimodular Pisot substitutions. Amer. J. Math. 128(5) (2006), 1219–1282.CrossRefGoogle Scholar
[Bow70]Bowen, R.. Markov partitions for Axiom A diffeomorphisms. Amer. J. Math. 92 (1970), 725–747.CrossRefGoogle Scholar
[Bru81]Bruijn, N. G.. Algebraic theory of Penrose’s non-periodic tilings of the plane i & ii. Non. Nederl. Akad. Wetensch. Proc. Ser. A 84 (1981), 38–66.Google Scholar
[BS07]Barge, M. and Swanson, R.. Rigidity in one-dimensional tiling spaces. Topology Appl. 154(17) (2007), 3095–3099.CrossRefGoogle Scholar
[BW72]Bowen, R. and Walters, P.. Expansive one-parameter flows. J. Differential Equations 12 (1972), 180–193.CrossRefGoogle Scholar
[Cla07]Clark, A.. The dynamics of tiling spaces. Open Problems in Topology, II. Ed. Pearl, Elliott. Elsevier B.V., Amsterdam, 2007, pp. 463–468.CrossRefGoogle Scholar
[CS06]Clark, A. and Sadun, L.. When shape matters: deformations of tiling spaces. Ergod. Th. & Dynam. Sys. 26(1) (2006), 69–86.CrossRefGoogle Scholar
[Dwo93]Dworkin, S.. Spectral theory and X-ray diffraction. J. Math. Phys. 34 (1993), 2965–2967.CrossRefGoogle Scholar
[Ell69]Ellis, R.. Lectures on Topological Dynamics. W. A. Benjamin, Inc., New York, 1969.Google Scholar
[GS87]Grünbaum, B. and Shephard, G. C.. Tilings and Patterns. W.H. Freeman, New York, 1987.Google Scholar
[Hof95]Hof, A.. On diffraction by aperiodic structures. Comm. Math. Phys. 169(169) (1995), 25–43.CrossRefGoogle Scholar
[Kat03]Katok, A.. Combinatorial Constructions in Ergodic Theory and Dynamics (University Lecture Series, 30). American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
[Kel95]Kellendonk, J.. Non-commutative geometry of tilings and gap labelling. Rev. Math. Phys. 7 (1995), 1133–1180.CrossRefGoogle Scholar
[Kel08]Kellendonk, J.. Pattern equivariant functions, deformations and equivalence of tiling spaces. Ergod. Th. & Dynam. Sys. 28 (2008), 1153–1176.CrossRefGoogle Scholar
[KH95]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[KP00]Kellendonk, J. and Putnam, I. F.. Tilings, C *-algebras, and K-theory. Directions in Mathematical Quasicrystals (CRM Monographics Series, 13). American Mathematical Society, Providence, RI, 2000, pp. 177–206.Google Scholar
[KS79]Keynes, H. B. and Sears, M.. ℱ-expansive transformation groups. General topology and its applications 10 (1979), 67–85.CrossRefGoogle Scholar
[Kwa10]Kwapisz, J.. Topological friction in aperiodic minimal ℝd-actions. Fund. Math. 207(2) (2010), 175–178.CrossRefGoogle Scholar
[LW03]Lagarias, J. C. and Wang, Y.. Substitution Delone sets. Discrete Comput. Geom. 29(2) (2003), 175–209.CrossRefGoogle Scholar
[Man79]Mane, R.. Expansive homeomorphisms and topological dimension. Trans. Amer. Math. Soc. 252 (1979), 313–319.CrossRefGoogle Scholar
[Mos92]Mossé, B.. Puissances de mots et reconnaissabilité des points fixes d’une substitution. Theoret. Comput. Sci. 99(2) (1992), 327–334.CrossRefGoogle Scholar
[Oka90]Oka, M.. Expansiveness of real flows. Tsukuba J. Math. 14(1) (1990), 1–8.CrossRefGoogle Scholar
[PFS]Priebe-Frank, N. and Sadun, L.. Topology of (some) tiling spaces without finite local complexity. Discrete Contin. Dyn. Syst. 23(3) (2009), 847–865.CrossRefGoogle Scholar
[Pri00]Priebe, N. M.. Towards a characterization of self-similar tilings in terms of derived Voronoi tessellations. Geom. Dedicata 79(3) (2000), 239–265.CrossRefGoogle Scholar
[PS01]Priebe, N. M. and Solomyak, B.. Characterization of planar pseudo-self-similar tilings. Discrete Comput. Geom. 26(3) (2001), 289–306.CrossRefGoogle Scholar
[Rad94]Radin, C.. The pinwheel tilings of the plane. Ann. of Math. (2) 139(3) (1994), 661–702.CrossRefGoogle Scholar
[Rad95]Radin, C.. Symmety and tilings. Notices Amer. Math. Soc. 42 (1995), 26–32.Google Scholar
[Rob96]Robinson, E. A. Jr. The dynamical properties of Penrose tilings. Trans. Amer. Math. Soc. 348(11) (1996), 4447–4464.CrossRefGoogle Scholar
[Rob04]Robinson, E. A.. Symbolic dynamics and tilings of ℝd. Symbolic Dynamics and its Applications (Proceedings of Symposia in Applied Mathematics, 60). American Mathematical Society, Providence, RI, 2004, pp. 81–119.CrossRefGoogle Scholar
[RW92]Radin, C. and Wolff, M.. Space tilings and local isomorphism. Geom. Dedicata 42 (1992), 335–360.CrossRefGoogle Scholar
[Sad98]Sadun, L.. Some generalizations of the pinwheel tiling. Discrete Comput. Geom. 20 (1998), 79–110.CrossRefGoogle Scholar
[Sad08]Sadun, L.. Topology of Tiling Spaces. American Mathematical Society, Providence, RI, 2008.CrossRefGoogle Scholar
[Sen95]Senechal, M.. Quasicrystals and Geometry. Cambridge University Press, Cambridge, 1995.Google Scholar
[Sol97]Solomyak, B.. Dynamics of self-similar tilings. Ergod. Th. & Dynam. Sys. 17(3) (1997), 695–738.CrossRefGoogle Scholar
[Sol98]Solomyak, B.. Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20(2) (1998), 265–279.CrossRefGoogle Scholar
[Sol06]Solomyak, B.. Tilings and dynamics. Lecture Notes, EMS Summer School on Combinatorics, Automata and Number Theory, Liege, Preprint, 8–19 May 2006.Google Scholar