Skip to main content
Log in

Weak Horseshoe with Bounded-Gap-Hitting Times

  • Published:
Communications in Mathematics and Statistics Aims and scope Submit manuscript

Abstract

In this paper, we consider weak horseshoe with bounded-gap-hitting times. For a flow \((M,\phi )\), it is shown that if the time one map \((M,\phi _1)\) has weak horseshoe with bounded-gap-hitting times, so is \((M,\phi _\tau )\) for all \(\tau \ne 0\). In addition, we prove that for an affine homeomorphism of a compact metric abelian group, positive topological entropy is equivalent to weak horseshoe with bounded-gap-hitting times.

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. Abramov, L.: On the entropy of a flow. Dok. Akad. Nauk. SSSR. 128, 873–875 (1959)

    MATH  Google Scholar 

  2. Adler, R., Konheim, A., McAndrew, M.: Topological entropy. Trans. Am. Math. Soc. 114, 309–319 (1965)

    Article  MathSciNet  Google Scholar 

  3. Bowen, R.: Entropy for group endomorphisms and homogeneous spaces. Trans. Am. Math. Soc. 153, 401–414 (1971)

    Article  MathSciNet  Google Scholar 

  4. Furstenberg, H.: Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton (1981)

    Book  Google Scholar 

  5. Huang, W., Li, J., Xu, L., Ye, X.: The existence of semi-horseshoes for partially hyperbolic diffeomorphisms (2019) (preprint)

  6. Huang, W., Li, H., Ye, X.: Family independence for topological and measurable dynamics. Trans. Am. Math. Soc. 364(10), 5209–5242 (2012)

    Article  MathSciNet  Google Scholar 

  7. Huang, W., Lu, K.: Entropy, chaos and weak horseshoe for infinite dimensional random dynamical systems. Commun. Pure Appl. Math. 70(10), 1987–2036 (2017)

    Article  MathSciNet  Google Scholar 

  8. Huang, W., Ye, X.: A local variational relation and applications. Israel J. Math. 151, 237–279 (2006)

    Article  MathSciNet  Google Scholar 

  9. Kerr, D., Li, H.: Independence in topological and \(C^*\)-dynamics. Math. Ann. 338(4), 869–926 (2007)

    Article  MathSciNet  Google Scholar 

  10. Ohno, T.: A weak equivalence and topological entropy. Publ. Res. Inst. Math. Sci. 16, 289–298 (1980)

    Article  MathSciNet  Google Scholar 

  11. Smale, S.: Diffeomorphisms with Many Periodic Points. 1965 Differential and Combinatorial Topology. In: A Symposium in Honor of Marston Morse. Princeton University Press, Princeton, pp. 63–80

  12. Sun, W., Young, T., Zhou, Y.: Topological entropies of equivalent smooth flows. Trans. Am. Math. Soc. 361, 3071–3082 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Leiye Xu is partially supported by NNSF of China (11801538, 11871188) and the USTC Research Funds of the Double First-Class Initiative. Junren Zheng is partially supported by NNSF of China (11971455). The authors would like to thank Wen Huang and Xiao Ma for making many valuable suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Leiye Xu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, L., Zheng, J. Weak Horseshoe with Bounded-Gap-Hitting Times. Commun. Math. Stat. 8, 463–472 (2020). https://doi.org/10.1007/s40304-020-00209-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40304-020-00209-4

Keywords

Mathematics Subject Classification

Navigation