Skip to main content
Log in

Polynomial Entropy for Interval Maps and Lap Number

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

We prove an upper bound for the polynomial entropy of continuous, piecewise monotone maps of the interval, according to the number of intervals of monotonicity of its iterates. We give examples that show that this inequality is sharp. As a direct consequence of this inequality, the polynomial entropy of monotone, continuous, interval maps is always less than or equal to one. We give examples where we can also obtain lower bounds. We also prove analogous inequality in terms of total variations of the iterates of these interval maps. Also, this inequality is sharp.

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

Data Availability Statement

All data generated or analysed during this study are included in this published article.

References

  1. Artigue, A., Carrasco-Oliveira, D., Monteverde, I.: Polynomial entropy and expansivity. Acta Math. Hung. 152, 140–149 (2017)

    Article  MathSciNet  Google Scholar 

  2. Baillif, M.: A polynomial bound for the lap number. Qual. Theory Dyn. Syst. 3, 325–329 (2002)

    Article  MathSciNet  Google Scholar 

  3. Bowen, R., Franks, J.: The periodic points of maps of the disk and the interval. Topology 15, 337–342 (1976)

    Article  MathSciNet  Google Scholar 

  4. Brucks, K.M., Bruin, H.: Topics from One-Dimensional Dynamics. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  5. Kozlovski, O., Shen, W., van Strien, S.: Density of hyperbolicity in dimension one. Ann. Math. 166, 145–182 (2007)

    Article  MathSciNet  Google Scholar 

  6. Labrousse, C.: Polynomial entropy for the circle homeomorphisms and for \(C^1\) nonvanishing vector fields on \({\mathbb{T}}^2\). arXiv: 1311.02131v1

  7. Marco, J.-P.: Polynomial entropies and integrable Hamiltonian systems. Regul. Chaotic Dyn. 18, 623–655 (2013)

    Article  MathSciNet  Google Scholar 

  8. Milnor, J., Thurston, W.: On iterated maps of the interval. Lect. Notes Math. 1342, 465–563 (2006)

    Article  MathSciNet  Google Scholar 

  9. Misiurewicz, M.: Structure of mappings of an interval with zero entropy. Publ. Math. IHES Paris 53, 5–16 (1981)

    Article  MathSciNet  Google Scholar 

  10. Misiurewicz, M., Szlenk, W.: Entropy of piecewise monotone mappings. Stud. Math. LXVII, 45–63 (1980)

    Article  MathSciNet  Google Scholar 

  11. Nitecki, Z.: Topological dynamics on the interval. In: Katok, A. (ed.) Ergodic Theory and Dynamical Systems II, pp. 1–73. Springer, New York (1982)

    Google Scholar 

  12. Rothschild, J.: On the computation of topological entropy. Thesis, CUNI (1971)

  13. Ruette, S.: Chaos on the Interval. University Lecture Series, vol. 67. American Mathematical Society, Providence (2017)

    Book  Google Scholar 

  14. Walters, P.: An Introduction to Ergodic Theory. Springer, Berlin (1982)

    Book  Google Scholar 

  15. Young, L.-S.: On the prevalence of horseshoes. Trans. Am. Math. Soc. 263, 75–88 (1981)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank the anonymous referee(s) for the careful reading and most invaluable suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to José Barbosa Gomes.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gomes, J.B., Carneiro, M.J.D. Polynomial Entropy for Interval Maps and Lap Number. Qual. Theory Dyn. Syst. 20, 21 (2021). https://doi.org/10.1007/s12346-021-00456-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-021-00456-y

Keywords

Mathematics Subject Classification

Navigation