Skip to main content
SpringerLink
Log in

Topological complexity, minimality and systems of order two on torus

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

Abstract

The dynamical system on \(\mathbb{T}^2\) which is a group extension over an irrational rotation on \(\mathbb{T}^1\) is investigated. The criterion when the extension is minimal, a system of order 2 and when the maximal equicontinuous factor is the irrational rotation is given. The topological complexity of the extension is computed, and a negative answer to the latter part of an open question raised by Host et al. (2014) is obtained.

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. Auslander J. Minimal Flows and Their Extensions. Amsterdam: Elsevier, 1988

    MATH  Google Scholar 

  2. Blanchard F, Host B, Maass A. Topological complexity. Ergodic Theory Dynam Syst, 2000, 20: 641–662

    Article  MathSciNet  MATH  Google Scholar 

  3. Dong P D, Donoso S, Maass A, et al. Infinite-step nilsystems, independence and topological complexity. Ergodic Theory Dynam Syst, 2013, 33: 118–143

    Article  MathSciNet  MATH  Google Scholar 

  4. Donoso S. Enveloping semigroups of systems of order d. Discrete Contin Dyn Syst, 2014, 34: 2729–2740

    Article  MathSciNet  MATH  Google Scholar 

  5. Einsiedler M, Ward T. Ergodic Theory-with a View towards Number Theory. Berlin: Springer-Verlag, 2011

    MATH  Google Scholar 

  6. Frączek K. Measure-preserving diffeomorphisms of the torus. Ergodic Theory Dynam Syst, 2001, 21: 1759–1781

    MathSciNet  MATH  Google Scholar 

  7. Frączek K. On difffeomorphisms with polynomial growth of the derivative on surfaces. Colloq Math, 2004, 99: 75–90

    Article  MathSciNet  MATH  Google Scholar 

  8. Furstenberg H. Strict ergodicity and transformation on the torus. Amer J Math, 1961, 83: 573–601

    Article  MathSciNet  MATH  Google Scholar 

  9. Glasner E. Minimal nil-transformations of class two. Israel J Math, 1993, 81: 31–51

    Article  MathSciNet  MATH  Google Scholar 

  10. Host B, Kra B, Maass A. Complexity of nilsystems and systems lacking nilfactors. J Anal Math, 2014, 124: 261–295

    Article  MathSciNet  MATH  Google Scholar 

  11. Oxtoby J C. Ergodic sets. Bull Amer Math Soc, 1952, 58: 116–136

    Article  MathSciNet  MATH  Google Scholar 

  12. Queffelec M. Substitution Dynamical Systems-Spectral Analysis. Heidelberg-Berlin: Springer-Verlag, 2010

    Book  MATH  Google Scholar 

  13. Walters P. An Introduction to Ergodic Theory. New York-Berlin: Springer-Verlag, 1982

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

    YiXiao Qiao

Authors
  1. YiXiao Qiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to YiXiao Qiao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qiao, Y. Topological complexity, minimality and systems of order two on torus. Sci. China Math. 59, 503–514 (2016). https://doi.org/10.1007/s11425-015-5042-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-015-5042-0

Keywords

MSC(2010)

Search

Navigation