Skip to main content
Log in

Topologically Mixing Maps and the Pseudoarc

  • Published:
Ukrainian Mathematical Journal Aims and scope

It is known that the pseudoarc can be constructed as the inverse limit of the copies of [0, 1] with one bonding map f which is topologically exact. On the other hand, the shift homeomorphism σ f is topologically mixing in this case. Thus, it is natural to ask whether f can be only mixing or must be exact. It has been recently observed that, in the case of some hereditarily indecomposable continua (e.g., pseudocircles) the property of mixing of a bonding map implies its exactness. The main aim of the present article is to show that the indicated kind of forcing of recurrence is not the case for the bonding map defining the pseudoarc.\

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

Access this article

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. M. Barge, H. Bruin, and S. Štimac, “The Ingram conjecture,” Geometry & Topology, 16, 2481–2516 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Barge and J. Martin, “Chaos, periodicity and snake-like continua,” Trans. Amer. Math. Soc., 289, 355–365 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  3. R. H. Bing, “A homogeneous indecomposable plane continuum,” Duke Math. J., 15, 729–742 (1948).

    Article  MathSciNet  MATH  Google Scholar 

  4. R. H. Bing, “Snake-like continua,” Duke Math. J., 18, 653–663 (1951).

    Article  MathSciNet  MATH  Google Scholar 

  5. R. H. Bing, “Concerning hereditarily indecomposable continua,” Pacif. J. Math., 1, 43–51 (1951).

    Article  MathSciNet  MATH  Google Scholar 

  6. L. Block, J. Keesling, and V. V. Uspenskij, “Inverse limits which are the pseudoarc,” Houston J. Math., 26, 629–638 (2000).

    MathSciNet  MATH  Google Scholar 

  7. C. Mouron, “Entropy of shift maps of the pseudo-arc,” Topology Appl., 159, 34–39 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Harańczyk, D. Kwietniak, and P. Oprocha, “Topological entropy and specification property for mixing graph maps,” Ergodic Theory & Dynam. Systems, publ. online, DOI:10.1017/etds.2013.6.

  9. G. W. Henderson, “The pseudo-arc as an inverse limit with one binding map,” Duke Math. J., 31, 421–425 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Kennedy, “Positive entropy homeomorphisms on the pseudoarc,” Mich. Math. J., 36, No. 2, 181–191 (1989).

    Article  MATH  Google Scholar 

  11. B. Knaster, “Un continu dont tout sous-continu est indécomposable,” Fund. Math., 3, 247–286 (1922).

    MATH  Google Scholar 

  12. P. Minc, and W. R. R. Transue, “A transitive map on [0, 1] whose inverse limit is the pseudoarc,” Proc. Amer. Math. Soc., 111, 1165–1170 (1991).

    MathSciNet  MATH  Google Scholar 

  13. E. E. Moise, “An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua,” Trans. Amer. Math. Soc., 63, 581–594 (1948).

    Article  MathSciNet  MATH  Google Scholar 

  14. P. Kościelniak and P. Oprocha, “Shadowing, entropy and a homeomorphism of the pseudoarc,” Proc. Amer. Math. Soc., 138, 1047–1057 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  15. P. Kościelniak, P. Oprocha, and M. Tuncali, “Hereditarily indecomposable inverse limits of graphs: shadowing, mixing and exactness,” Proc. Amer. Math. Soc., 142, 681–694 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  16. X. Ye, “Topological entropy of the induced maps of the inverse limits with bonding maps,” Topol. Appl., 67, 113–118 (1995).

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 2, pp. 176–186, February, 2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Drwiega, T., Oprocha, P. Topologically Mixing Maps and the Pseudoarc. Ukr Math J 66, 197–208 (2014). https://doi.org/10.1007/s11253-014-0922-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-014-0922-y

Keywords

Navigation