Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T20:03:51.416Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact regularity and the cohomology of tiling spaces

Published online by Cambridge University Press:  18 January 2011

LORENZO SADUN
LORENZO SADUN*
Affiliation:
Department of Mathematics, University of Texas, Austin, TX 78712, USA (email: sadun@math.utexas.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

Exact regularity was introduced recently as a property of homological Pisot substitutions in one dimension. In this paper, we consider the analog of exact regularity for arbitrary tiling spaces. Let T be a d-dimensional repetitive tiling, and let Ω be its hull. If Ȟd(Ω,ℚ)=ℚk, then there exist k patches each of whose appearances governs the number of appearances of every other patch. This gives uniform estimates on the convergence of all patch frequencies to the ergodic limit. If the tiling T comes from a substitution, then we can quantify that convergence rate. If T is also one dimensional, we put constraints on the measure of any cylinder set in Ω.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 6 , December 2011 , pp. 1819 - 1834
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[AP]Anderson, J. E. and Putnam, I. F.. Topological invariants for substitution tilings and their associated C *-algebras. Ergod. Th. & Dynam. Sys. 18 (1998), 509–537.Google Scholar
[BBG]Bellissard, J., Benedetti, R. and Gambaudo, J.-M.. Spaces of tilings, finite telescopic approximations and gap-labelling. Comm. Math. Phys. 261 (2006), 1–41.CrossRefGoogle Scholar
[BBJS]Barge, M., Bruin, H., Jones, L. and Sadun, L.. Homological Pisot substitutions and exact regularity. Available at arXiv:1001.2027v1.Google Scholar
[BD]Barge, M. and Diamond, B.. Cohomology in one-dimensional substitution tiling spaces. Proc. Amer. Math. Soc. 136(6) (2008), 2183–2191.CrossRefGoogle Scholar
[BDHS]Barge, M., Diamond, B., Hunton, J. and Sadun, L.. Cohomology of substitution tiling spaces. Ergod. Th. & Dynam. Sys. to appear. Available at arXiv:0811.2507.Google Scholar
[CGU]Chazottes, J.-R., Gambaudo, J.-M. and Ugalde, E.. On the geometry of ground states and quasicrystals in lattice systems. Preprint, arXiv:0802.3661.Google Scholar
[CS]Clark, A. and Sadun, L.. When shape matters: deformations of tiling spaces. Ergod. Th. & Dynam. Sys. 26 (2006), 69–86.CrossRefGoogle Scholar
[Kal]Kalahurka, W.. Rotational cohomology and total pattern equivariant cohomology of tiling spaces acted on by infinite groups. PhD Thesis, Department of Mathematics, University of Texas, 2010.Google Scholar
[Kel1]Kellendonk, J.. Non-commutative geometry of tilings and gap-labeling. Rev. Math. Phys. 7 (1995), 1133–1180.CrossRefGoogle Scholar
[Kel2]Kellendonk, J.. Pattern-equivariant functions and cohomology. J. Phys. A 36 (2003), 1–8.CrossRefGoogle Scholar
[KP]Kellendonk, J. and Putnam, I.. The Ruelle–Sullivan map for ℝn-actions. Math. Ann. 334 (2006), 693–711.Google Scholar
[Po]Pohst, M.. A note on index divisors. Computational Number Theory. Eds. Pehto, A., Pohst, M., Williams, H. and Zimmer, H.. de Gruyter, Berlin, 1991, pp. 173–182.Google Scholar
[Rad]Radin, C.. The pinwheel tilings of the plane. Ann. of Math. (2) 139 (1994), 661–702.Google Scholar
[Ran]Rand, B.. Pattern-equivariant cohomology of tiling spaces with rotations. PhD Thesis, Department of Mathematics, University of Texas, 2006.Google Scholar
[Sad]Sadun, L.. Pattern-equivariant cohomology with integer coefficients. Ergod. Th. & Dynam. Sys. 27 (2007), 1991–1998.Google Scholar
[WT]Taixes, X., Ventosa, I. and Wiese, G.. Computing congruences of modular forms and Galois representations modulo prime powers. Preprint, arXiv:0909.2724v2.Google Scholar