Skip to main content
SpringerLink
Log in

Topologically conjugate classifications of the translation actions on low-dimensional compact connected Lie groups

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

In this article, we focus on the left translation actions on noncommutative compact connected Lie groups with topological dimension 3 or 4, consisting of SU(2), U(2), SO(3), SO(3) × S1 and Spinℂ(3). We define the rotation vectors (numbers) of the left actions induced by the elements in the maximal tori of these groups, and utilize rotation vectors (numbers) to give the topologically conjugate classification of the left actions. Algebraic conjugacy and smooth conjugacy are also considered. As a by-product, we show that for any homeomorphism f : L(p, −1) × S1 → L(p, −1) × S1, the induced isomorphism (π ◦ f ◦ i)* maps each element in the fundamental group of L(p, −1) to itself or its inverse, where i : L(p, −1) → L(p, −1) × S1 is the natural inclusion and π : L(p, −1) × S1 → L(p, −1) is the projection.

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. Adler R L, Palais R. Homeomorphic conjugacy of automorphisms on the torus. Proc Amer Math Soc, 1965, 16: 1222–1225

    Article  MathSciNet  MATH  Google Scholar 

  2. Adler R L, Tresser C, Worfolk P. Topological conjugacy of linear endomorphisms of the 2-torus. Trans Amer Math Soc, 1997, 394: 1633–1652

    Article  MathSciNet  MATH  Google Scholar 

  3. Bhattacharya S. Orbit equivalence and topological conjugacy of affine actions on compact abelian groups. Monatsh Math, 2000, 129: 89–96

    Article  MathSciNet  MATH  Google Scholar 

  4. Bhattacharya S. Topological conjugacy of automorphism flows on compact Lie groups. Ergodic Theory Dynam Syst, 2000, 20: 335–342

    Article  MathSciNet  MATH  Google Scholar 

  5. Connes A. A new proof of Morley’s theorem. Publ Math Inst Hautes Etudes Sci, 1998, S88: 43–46

    MathSciNet  MATH  Google Scholar 

  6. Dávalos P. On torus homeomorphisms whose rotation set is an interval. Math Z, 2013, 275: 1005–1045

    Article  MathSciNet  MATH  Google Scholar 

  7. Elliott G. The classification problem for amenable C*-algebras. In: Proceedings of the International Congress of Mathematicians. Basel: Birkhäuser, 1994, 922–932

    Google Scholar 

  8. Elliott G, Gong G. On inductive limits of matrix algebras over the two-torus. Amer J Math, 1996, 118: 263–290

    Article  MathSciNet  MATH  Google Scholar 

  9. Elliott G, Gong G. On the classification of C*-algebras of real rank zero II. Ann of Math (2), 1996, 144: 497–610

    Article  MathSciNet  MATH  Google Scholar 

  10. Elliott G, Gong G, Li L. On the classification of simple inductive limit C*-algebras II: The isomorphism theorem. Invent Math, 2007, 168: 249–320

    Article  MathSciNet  MATH  Google Scholar 

  11. Elliott G, Ivanescu C. The classification of separable simple C*-algebras which are inductive limits of continuous-trace C*-algebras with spectrum homeomorphic to the closed interval [0,1]. J Funct Anal, 2007, 254: 879–903

    Article  MathSciNet  MATH  Google Scholar 

  12. Elliott G, Li H. Morita equivalence of smooth noncommutative tori. Acta Math, 2007, 199: 1–27

    Article  MathSciNet  MATH  Google Scholar 

  13. Elliott G, Li H. Strong Morita equivalence of higher-dimensional noncommutative tori II. Math Ann, 2008, 341: 825–844

    Article  MathSciNet  MATH  Google Scholar 

  14. Franks J. Anosov diffeomorphism. In: Global Analysis. Proceedings of Symposia in Pure Mathematics, vol. 14. Providence: Amer Math Soc, 1970, 61–93

    Article  MathSciNet  Google Scholar 

  15. Franks J. Realizing rotation vectors for torus homeomorphisms. Trans Amer Math Soc, 1989, 311: 107–115

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Gong G. Classification of C*-algebras of real rank zero and unsuspended E-equivalence types. J Funct Anal, 1998, 152: 281–329

    Article  MathSciNet  MATH  Google Scholar 

  18. Hatcher A. Notes on basic 3-manifold topology. Topology, 2000, 138: 2244–2247

    Google Scholar 

  19. Hou B, Liu H, Pan X. Mutual embeddability equivalence relation for rotation algebras. J Math Anal Appl, 2017, 452: 495–504

    Article  MathSciNet  MATH  Google Scholar 

  20. Husemoller D. Fibre Bundles. Berlin-Heidelberg-New York: Springer, 1966

    Book  MATH  Google Scholar 

  21. Koropecki A. On the dynamics of torus homeomorphisms. PhD Thesis. Rio de Janeiro: Instituto de Matematica Pura e Aplicada, 2008

    MATH  Google Scholar 

  22. Kuiper N H, Robbin J W. Topological classification of linear endomorphisms. Invent Math, 1973, 19: 83–106

    Article  MathSciNet  MATH  Google Scholar 

  23. Kwasik S, Schultz R. On s-cobordisms of metacyclic prism manifolds. Invent Math, 1989, 97: 526–552

    Article  MathSciNet  MATH  Google Scholar 

  24. Li H. Strong Morita equivalence of higher-dimensional noncommutative tori. J Reine Angew Math, 2004, 576: 167–180

    MathSciNet  MATH  Google Scholar 

  25. Lin H. Classification of simple C*-algebras and higher dimensional noncommutative tori. Ann of Math (2), 2003, 157: 521–544

    Article  MathSciNet  MATH  Google Scholar 

  26. Lin H. Classification of simple C*-algebras of tracial topological rank zero. Duke Math J, 2004, 125: 91–119

    Article  MathSciNet  MATH  Google Scholar 

  27. Lin H. Furstenberg transformations and approximate conjugacy. Canad J Math, 2008, 60: 189–207

    Article  MathSciNet  MATH  Google Scholar 

  28. Lin H. On Locally AH Algebras. Memoris of the American Mathematical Society, vol. 235. Providence: Amer Math Soc, 2015

  29. Lin H. Crossed products and minimal dynamical systems. J Topol Anal, 2018, 10: 447–469

    Article  MathSciNet  MATH  Google Scholar 

  30. Lin H, Phillips N C. Crossed products by minimal homeomorphisms. J Reine Angew Math, 2010, 641: 95–122

    MathSciNet  MATH  Google Scholar 

  31. Liu H. Smooth crossed product of minimal unique ergodic diffeomorphism of odd sphere. J Noncommut Geom, 2017, 11: 1381–1393

    Article  MathSciNet  MATH  Google Scholar 

  32. Liu H. Smooth crossed product of minimal unique ergodic diffeomorphisms of a manifold and cyclic cohomology. J Topol Anal, 2019, 11: 739–751

    Article  MathSciNet  MATH  Google Scholar 

  33. Manning A. There are no new Anosov diffeomorphisms on tori. Amer J Math, 1974, 96: 422–429

    Article  MathSciNet  MATH  Google Scholar 

  34. Pimsner M, Voiculescu D. Exact sequences for K-groups and Ext-groups of certain cross-product C*-algebras. J Operator Theory, 1980, 4: 93–118

    MathSciNet  MATH  Google Scholar 

  35. Price J F. Lie Groups and Compact Groups. Cambridge: Cambridge University Press, 1977

    Book  MATH  Google Scholar 

  36. Rieffel M. C*-algebras associated with irrational rotations. Pacific J Math, 1981, 93: 415–429

    Article  MathSciNet  MATH  Google Scholar 

  37. Robbin J W. Topological conjugacy and structural stability for discrete dynamical systems. Bull Amer Math Soc, 1972, 78: 923–952

    Article  MathSciNet  MATH  Google Scholar 

  38. Schultz R. On the topological classification of linear representations. Topology, 1977, 16: 263–269

    Article  MathSciNet  MATH  Google Scholar 

  39. Sepanski M R. Compact Lie Groups. New York: Springer, 2007

    Book  MATH  Google Scholar 

  40. Smale S. Dynamical systems and the topological conjugacy problem for diffeomorphisms. In: Proceedings of the International Congress of Mathematicians. Djursholm: Inst Mittag-Leffler, 1963, 490–496

    Google Scholar 

  41. Sun H. Degree ±1 self-maps and self-homeomorphisms on prime 3-manifolds. Algebr Geom Topol, 2010, 10: 867–890

    Article  MathSciNet  MATH  Google Scholar 

  42. Swanson R, Walker R. Boundaries of rotation sets for homeomorphisms of the n-torus. Proc Amer Math Soc, 1996, 124: 3247–3255

    Article  MathSciNet  MATH  Google Scholar 

  43. Tal F. Transitivity and rotation sets with nonempty interior for homeomorphisms of the 2-torus. Proc Amer Math Soc, 2012, 140: 3567–3579

    Article  MathSciNet  MATH  Google Scholar 

  44. Tomiyama J. Topological full groups and structure of normalizers in transformation group C*-algebras. Pacific J Math, 1996, 173: 571–583

    Article  MathSciNet  MATH  Google Scholar 

  45. Toms A, Winter W. Minimal dynamics and the classification of C*-algebras. Proc Natl Acad Sci USA, 2009, 106: 16942–16943

    Article  MathSciNet  MATH  Google Scholar 

  46. Walters P. Topological conjugacy of affine transformations of tori. Trans Amer Math Soc, 1968, 131: 40–50

    Article  MathSciNet  MATH  Google Scholar 

  47. Walters P. Topological conjugacy of affine transformations of compact abelian groups. Trans Amer Math Soc, 1969, 140: 95–107

    Article  MathSciNet  MATH  Google Scholar 

  48. Weinberger S. Some remarks inspired by the C 0 Zimmer program. In: Geometry, Rigidity, and Group Actions. Chicago Lectures in Mathematics. Chicago: University of Chicago Press, 2011, 262–282

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, 130012, China

    Xiaotian Pan & Bingzhe Hou

Authors
  1. Xiaotian Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bingzhe Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bingzhe Hou.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pan, X., Hou, B. Topologically conjugate classifications of the translation actions on low-dimensional compact connected Lie groups. Sci. China Math. 64, 963–1010 (2021). https://doi.org/10.1007/s11425-019-9535-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-019-9535-x

Keywords

MSC(2010)

Search

Navigation