Skip to main content
SpringerLink
Log in

Full groups of Cantor minimal systems

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We associate different types of full groups to Cantor minimal systems. We show how these various groups (as abstract groups) are complete invariants for orbit equivalence, strong orbit equivalence and flip conjugacy, respectively. Furthermore, we introduce a group homomorphism, the socalled mod map, from the normalizers of the various full groups to the automorphism groups of the (ordered)K 0-groups, which are associated to the Cantor minimal systems. We show how this in turn is related to the automorphisms of the associatedC *-crossed products. Our results are analogues in the topological dynamical setting of results obtained by Dye, Connes-Krieger and Hamachi-Osikawa in measurable dynamics.

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

  • [B1] M. Boyle,Topological orbit equivalence and factor maps in symbolic dynamics, Ph.D. Thesis, University of Washington, 1983.

  • [B2] M. Boyle,A homeomorphism good on measures and bad on orbits, preprint.

  • [BT] M. Boyle and J. Tomiyama,Bounded topological orbit equivalence and C *-algebras, preprint.

  • [C] A. Connes,Non-commutative differential geometry, Publications Mathématiques de l’Institut des Hautes Études Scientifiques62 (1986), 41–144.

    Article  Google Scholar 

  • [CK] A. Connes and W. Krieger,Measure space automorphisms, the normalizers of their full groups, and approximate finiteness, Journal of Functional Analysis18 (1975), 318–327.

    Article  MATH  Google Scholar 

  • [CT] A. Connes and M. Takesaki,The flow of weights on factors of type III, Tôhoku Mathematical Journal29 (1977), 473–575.

    Article  MATH  MathSciNet  Google Scholar 

  • [D1] H. Dye,On groups of measure preserving transformations I, American Journal of Mathematics81 (1959), 119–159.

    Article  MATH  MathSciNet  Google Scholar 

  • [D2] H. Dye,On groups of measure preserving transformations II, American Journal of Mathematics85 (1963), 551–576.

    Article  MATH  MathSciNet  Google Scholar 

  • [E] E. G. Effros,Dimensions and C *-algebras, Conf. Board Math. Sci.,46, American Mathematical Society, Providence, R.I., 1981.

    MATH  Google Scholar 

  • [GPS] T. Giordano, I. F. Putnam and C. F. Skau,Topological orbit equivalence and C *-crossed products, Journal für die reine und angewandte Mathematik469 (1995), 51–111.

    MATH  MathSciNet  Google Scholar 

  • [GW] E. Glasner and B. Weiss,Weak orbit equivalence of Cantor minimal systems, International Journal of Mathematics23 (1996), 737–769.

    Google Scholar 

  • [GW1] E. Glasner and B. Weiss,On the construction of minimal skew products, Israel Journal of Mathematics34 (1979), 321–336.

    Article  MATH  MathSciNet  Google Scholar 

  • [H] P. R. Halmos,Lectures on Boolean Algebras, Van Nostrand Mathematical Studies 1, D. Van Nostrand Company, Princeton, 1967.

    Google Scholar 

  • [HO] T. Hamachi and M. Osikawa,Fundamental Homomorphism of Normalizer of Ergodic Transformation, Lecture Notes in Mathematics729, Springer-Verlag, Berlin, 1978.

    Google Scholar 

  • [HPS] R. H. Herman, I. F. Putnam and C. F. Skau,Ordered Bratteli diagrams, dimension groups and topological dynamics, International Journal of Mathematics3 (1992), 827–864.

    Article  MATH  MathSciNet  Google Scholar 

  • [J] Ø. Johansen,Cantor systems, dimension groups and Bratteli diagrams, Ms. Thesis, University of Trondheim-AVH, 1994.

  • [K1] W. Krieger,On ergodic flows and isomorphisms of factors, Mathematische Annalen223 (1976), 19–70.

    Article  MATH  MathSciNet  Google Scholar 

  • [K2] W. Krieger,On a dimension for a class of homeomorphism groups, Mathematische Annalen252 (1980), 87–95.

    Article  MATH  MathSciNet  Google Scholar 

  • [Ku] K. Kuratowski,Topology, Vol. II, Academic Press, New York, 1968.

    Google Scholar 

  • [Kup] I. Kupka,On two notions of structural stability, Journal of Differential Geometry9 (1974), 639–644.

    MATH  MathSciNet  Google Scholar 

  • [P] I. F. Putnam,The C *-algebras associated with minimal homeomorphisms of the Cantor set, Pacific Journal of Mathematics136 (1989), 329–353.

    MATH  MathSciNet  Google Scholar 

  • [R] J. Renault,A groupoid approach to C *-algebras, Lecture Notes in Mathematics793, Springer-Verlag, Berlin, 1980.

    MATH  Google Scholar 

  • [SV] Ş. Strâtilâ and D. Voiculescu,Representations of AF-Algebras and of the Group U(∞), Lecture Notes in Mathematics486, Springer-Verlag, Berlin, 1975.

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Thierry Giordano

  2. Department of Mathematics and Statistics, University of Victoria, V8W 3P4, Victoria, B.C., Canada

    Ian F. Putnam

  3. Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), N-7034, Trondheim, Norway

    Christian F. Skau

Authors
  1. Thierry Giordano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ian F. Putnam
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christian F. Skau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thierry Giordano.

Additional information

Research supported in part by operating grants from NSERC (Canada).

Research supported in part by the Norwegian Research Council for Science and Humanities.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Giordano, T., Putnam, I.F. & Skau, C.F. Full groups of Cantor minimal systems. Isr. J. Math. 111, 285–320 (1999). https://doi.org/10.1007/BF02810689

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02810689

Keywords

Search

Navigation