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Pattern equivariant functions, deformations and equivalence of tiling spaces

Published online by Cambridge University Press:  01 August 2008

JOHANNES KELLENDONK
JOHANNES KELLENDONK*
Affiliation:
Université de Lyon, Université Lyon 1, CNRS, UMR 5208 Institut Camille Jordan, Batiment du Doyen Jean Braconnier, 43 blvd du 11 novembre 1918, F-69622 Villeurbanne Cedex, France
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Abstract

We re-investigate the theory of deformations of tilings using P-equivariant cohomology. In particular, we relate the notion of asymptotically negligible shape functions introduced by Clark and Sadun to weakly P-equivariant forms. We then investigate more closely the relation between deformations of patterns and homeomorphism or topological conjugacy of pattern spaces.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 4 , August 2008 , pp. 1153 - 1176
Copyright
Copyright © 2008 Cambridge University Press

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References

[1]Baake, M., Schlottmann, M. and Jarvis, P. D.. Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability. J. Phys. A: Math. Gen. 24 (1991), 4637–4654.CrossRefGoogle Scholar
[2]Clark, A. and Sadun, L.. When shape matters: deformations of tiling spaces. Ergod. Th. & Dynam. Sys. 26 (2006), 69–86.CrossRefGoogle Scholar
[3]Gähler, F.. Talk given at the conference ‘Aperiodic Order, Dynamical Systems, Operator Algebras and Topology’ 2002, slides available at: www.pims.math.ca/science/2002/adot/lectnotes/Gaehler.Google Scholar
[4]Kellendonk, J.. Topological equivalence of tilings. J. Math. Phys. 38 (1997), 1823–1842.CrossRefGoogle Scholar
[5]Kellendonk, J.. Pattern-equivariant functions and cohomology. J. Phys. A: Math. Gen. 36 (2003), 1–8.CrossRefGoogle Scholar
[6]Kellendonk, J. and Richard, S.. Topological boundary maps in physics: general theory and applications. Perspectives in Operator Algebras and Mathematical Physics. Theta, Bucharest, 2008, pp. 105–121.Google Scholar
[7]Kellendonk, J. and Putnam, I. F.. The Ruelle–Sullivan map for actions of . Math. Ann. 334 (2006), 693–711.CrossRefGoogle Scholar
[8]Sadun, L.. Tiling spaces are inverse limits. J. Math. Phys. 44 (2003), 5410–5414.CrossRefGoogle Scholar
[9]Sadun, L.. Pattern-equivariant cohomology with integer coefficients. Ergod. Th. & Dynam. Sys. 27(6) (2007), 1991–1998.CrossRefGoogle Scholar
[10]Sadun, L. and Williams, R. F.. Tiling spaces are Cantor set fiber bundles. Ergod. Th. & Dynam. Sys. 23 (2003), 307–316.CrossRefGoogle Scholar