SpringerLink
Log in

Orbit equivalence for Cantor minimal ℤd-systems

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We show that every minimal action of any finitely generated abelian group on the Cantor set is (topologically) orbit equivalent to an AF relation. As a consequence, this extends the classification up to orbit equivalence of minimal dynamical systems on the Cantor set to include AF relations and ℤd-actions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bellissard, J., Benedetti, R., Gambaudo, J.-M.: Spaces of tilings, finite telescopic approximations and gap-labeling. Commun. Math. Phys. 261, 1–41 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Benedetti, R., Gambaudo, J.-M.: On the dynamics of G-solenoids. Applications to Delone sets. Ergod. Theory Dyn. Syst. 23, 673–691 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cheng, S.-W., Dey, T.K., Edelsbrunner, H., Facello, M.A., Teng, S.-H.: Sliver exudation. J. ACM 47, 883–904 (2000)

    Article  MathSciNet  Google Scholar 

  4. Connes, A., Feldman, J., Weiss, B.: An amenable equivalence relation is generated by a single transformation. Ergod. Theory Dyn. Syst. 1, 431–450 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dougherty, R., Jackson, S., Kechris, A.S.: The structure of hyperfinite Borel equivalence relations. Trans. Am. Math. Soc. 341, 193–225 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dye, H.A.: On groups of measure preserving transformation, I. Am. J. Math. 81, 119–159 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  7. Forrest, A.: A Bratteli diagram for commuting homeomorphisms of the Cantor set. Int. J. Math. 11, 177–200 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gao, S., Jackson, S.: Countable abelian group actions and hyperfinite equivalence relations. Preprint

  9. Giordano, T., Matui, H., Putnam, I.F., Skau, C.F.: Orbit equivalence for Cantor minimal ℤ2-systems. J. Am. Math. Soc. 21, 863–892 (2008). math.DS/0609668

    Article  MathSciNet  Google Scholar 

  10. Giordano, T., Matui, H., Putnam, I.F., Skau, C.F.: The absorption theorem for affable equivalence relations. Ergod. Theory Dyn. Syst. 28, 1509–1531 (2008). ar**v:0705.3270

    Article  MATH  MathSciNet  Google Scholar 

  11. Giordano, T., Putnam, I.F., Skau, C.F.: Topological orbit equivalence and C *-crossed products. J. Reine Angew. Math. 469, 51–111 (1995)

    MATH  MathSciNet  Google Scholar 

  12. Giordano, T., Putnam, I.F., Skau, C.F.: Affable equivalence relations and orbit structure of Cantor dynamical systems. Ergod. Theory Dyn. Syst. 24, 441–475 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Giordano, T., Putnam, I.F., Skau, C.F.: Cocycles for Cantor minimal ℤd-systems, Int. J. Math. (2009, to appear)

  14. Herman, R.H., Putnam, I.F., Skau, C.F.: Ordered Bratteli diagrams, dimension groups and topological dynamics. Int. J. Math. 3, 827–864 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jackson, S., Kechris, A.S., Louveau, A.: Countable Borel equivalence relations. J. Math. Log. 2, 1–80 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Johansen, Ø.: Ordered K-theory and Bratteli diagrams: Implications for Cantor minimal systems. Ph.D. thesis, NTNU (1998)

  17. Kechris, A.S., Miller, B.D.: Topics in Orbit Equivalence. Lecture Notes in Math., vol. 1852. Springer, Berlin (2004)

    MATH  Google Scholar 

  18. Kellendonk, J., Putnam, I.F.: Tilings, C *-algebras, and K-theory. In: Directions in Mathematical Quasicrystals. CRM Monogr. Ser., vol. 13, pp. 177–206. Am. Math. Soc., Providence (2000)

    Google Scholar 

  19. Krieger, W.: On ergodic flows and the isomorphism of factors. Math. Ann. 223, 19–70 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  20. Mackey, G.W.: Ergodic theory and virtual groups. Math. Ann. 166, 187–207 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  21. Matous̆ek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer, New York (2002)

    Google Scholar 

  22. Matui, H.: Affability of equivalence relations arising from two-dimensional substitution tilings. Ergod. Theory Dyn. Syst. 26, 467–480 (2006). math.DS/0506251

    Article  MATH  MathSciNet  Google Scholar 

  23. Matui, H.: A short proof of affability for certain Cantor minimal ℤ2-systems. Can. Math. Bull. 50, 418–426 (2007). math.DS/0506250

    MATH  MathSciNet  Google Scholar 

  24. Matui, H.: An absorption theorem for minimal AF equivalence relations on Cantor sets. J. Math. Soc. Jpn. 60, 1171–1185 (2008). ar**v:0712.0733

    Article  MATH  MathSciNet  Google Scholar 

  25. Ornstein, D.S., Weiss, B.: Ergodic theory of amenable group actions I: The Rohlin lemma. Bull. Am. Math. Soc. 2, 161–164 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  26. Putnam, I.F.: Orbit equivalence of Cantor minimal systems: a survey and a new proof. Expo. Math. (2009, accepted)

  27. Renault, J.: A Groupoid Approach to C *-algebras. Lecture Notes in Mathematics, vol. 793. Springer, Berlin (1980)

    Google Scholar 

  28. Schlottmann, M.: Periodic and quasi-periodic Laguerre tilings. Int. J. Mod. Phys. B 7, 1351–1363 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  29. Stanley, R.P.: Decompositions of rational convex polytopes. Ann. Discrete Math. 6, 333–342 (1980)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, Ontario, Canada, K1N 6N5

    Thierry Giordano

  2. Graduate School of Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan

    Hiroki Matui

  3. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada, V8W 3R4

    Ian F. Putnam

  4. Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491, Trondheim, Norway

    Christian F. Skau

Authors
  1. Thierry Giordano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hiroki Matui
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ian F. Putnam
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Christian F. Skau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hiroki Matui.

Additional information

T. Giordano is supported in part by a grant from NSERC, Canada.

H. Matui is supported in part by a grant from the Japan Society for the Promotion of Science.

I.F. Putnam is supported in part by a grant from NSERC, Canada.

C.F. Skau is supported in part by the Norwegian Research Council.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Giordano, T., Matui, H., Putnam, I.F. et al. Orbit equivalence for Cantor minimal ℤd-systems. Invent. math. 179, 119–158 (2010). https://doi.org/10.1007/s00222-009-0213-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-009-0213-7

Keywords

Search

Navigation