Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-28T09:10:12.591Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometric realization for substitution tilings

Published online by Cambridge University Press:  05 December 2012

MARCY BARGE  and
JEAN-MARC GAMBAUDO
MARCY BARGE
Affiliation:
Montana State University, Bozeman, MT, 59717, USA (email: [email protected])
JEAN-MARC GAMBAUDO
Affiliation:
INLN, UMR CNRS 7335, Université de Nice-Sophia Antipolis 1361, route des Lucioles, 06560 Valbonne, France (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Given an n-dimensional substitution Φ whose associated linear expansion Λ is unimodular and hyperbolic, we use elements of the one-dimensional integer Čech cohomology of the tiling space ΩΦ to construct a finite-to-one semi-conjugacy G:ΩΦ→𝕋D, called a geometric realization, between the substitution induced dynamics and an invariant set of a hyperbolic toral automorphism. If Λ satisfies a Pisot family condition and the rank of the module of generalized return vectors equals the generalized degree of Λ, G is surjective and coincides with the map onto the maximal equicontinuous factor of the ℝn-action on ΩΦ. We are led to formulate a higher-dimensional generalization of the Pisot substitution conjecture: if Λ satisfies the Pisot family condition and the rank of the one-dimensional cohomology of ΩΦ equals the generalized degree of Λ, then the ℝn-action on ΩΦhas pure discrete spectrum.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 2 , April 2014 , pp. 457 - 482
Copyright
©2012 Cambridge University Press 

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

[A]Auslander, J.. Minimal Flows and their Extensions (North-Holland Mathematical Studies, 153). Elsevier, North-Holland, Amsterdam, New York, Oxford, and Tokyo, 1988.Google Scholar
[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
[BBJS]Barge, M., Bruin, H., Jones, L. and Sadun, L.. Homological Pisot substitutions and exact regularity. Israel J. Math. (2012), 1–20.Google Scholar
[BD1]Barge, M. and Diamond, B.. Cohomology in one-dimensional substitution tiling spaces. Proc. Amer. Math. Soc. 136(6) (2008), 2183–2191.Google Scholar
[BK]Barge, M. and Kellendonk, J.. Proximality and pure point spectrum for tiling dynamical systems. Preprint, 2011.Google Scholar
[BKw]Barge, M. and Kwapisz, J.. Hyperbolic pseudo-Anosov maps a.e. embed into toral automorphisms. Ergod. Th. & Dynam. Sys. 26 (2006), 961–972.Google Scholar
[BO]Barge, M. and Olimb, C.. Asymptotic structure in substitution tiling spaces. Preprint, 2010.Google Scholar
[BSW]Barge, M., Štimac, S. and Williams, R. F.. Pure discrete spectrum in substitution tiling spaces. Preprint, 2011.Google Scholar
[B]Berger, R.. The Undecidability of the Domino Problem (Memoirs of the American Mathematical Society, 66). American Mathematical Society, Providence, RI, 1966, pp. 1–72.Google Scholar
[BS]Berthé, V. and Siegel, A.. Tilings associated with beta-numeration and substitutions. Integers 5(3) (2005), A2, 46.Google Scholar
[F]Fathi, A.. Some compact invariant sets for hyperbolic linear automorphisms of torii. Ergod. Th. & Dynam. Sys. 8 (1988), 191–204.Google Scholar
[Fr]Franks, J.. Anosov diffeomorphisms. Global Analysis (Berkeley, CA, 1968) (Proceedings of Symposia Pure Mathematics, 14). American Mathematical Society, Providence, RI, 1970, pp. 61–93.Google Scholar
[HS]Hollander, M. and Solomyak, B.. Two-symbol Pisot substitutions have pure discrete spectrum. Ergod. Th. & Dynam. Sys. 23 (2003), 533–540.Google Scholar
[K]Kwapisz, J.. Personal communication.Google Scholar
[K1]Kenyon, R.. The construction of self-similar tilings. Geom. Funct. Anal. 6(3) (1996), 471–488.Google Scholar
[KS]Kenyon, R. and Solomyak, B.. On the characterization of expansion maps for self-affine tilings. Discrete Comput. Geom. 43 (2010), 577–593.Google Scholar
[LS1]Lee, J.-Y. and Solomyak, B.. Pisot family self-affine tilings, discrete spectrum, and the Meyer property. Preprint, 2010.Google Scholar
[LS2]Lee, J.-Y. and Solomyak, B.. Pure point diffractive substitution Delone sets have the Meyer property. Discrete Comput. Geom. 34 (2008), 319–338.Google Scholar
[M]Moody, R.. Meyer sets and their duals. The Mathematics of Long-Range Aperiodic Order. Ed. Moody, R.. Kluwer Academic Publishers, Dordrecht, 1997, pp. 403–441.Google Scholar
[P]Putnam, I. F.. A homology theory for Smale spaces. Preprint, 2010.Google Scholar
[R]Robinson, E. A.. A Halmos–von Neumann theorem for model sets, and almost automorphic dynamical systems. Dynamics, Ergodic Theory, and Geometry (Mathematical Sciences Research Institute Publications, 54). Cambridge University Press, Cambridge, 2007, pp. 243–272.Google Scholar
[S1]Solomyak, B.. Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20 (1998), 265–279.CrossRefGoogle Scholar
[S2]Solomyak, B.. Eigenfunctions for substitution tiling systems. Adv. Stud. Pure Math. 49 (2007), 433–454.Google Scholar
[V]Veech, W. A.. The equicontinuous structure relation for minimal Abelian transformation groups. Amer. J. Math. 90 (1968), 723–732.Google Scholar