Skip to main content
SpringerLink
Log in

Chaos and Entropy for Interval Maps

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

In this paper, various chaotic properties and their relationships for interval maps are discussed. It is shown that the proximal relation is an equivalence relation for any zero entropy interval map. The structure of the set of f-nonseparable pairs is well demonstrated and so is its relationship to Li-Yorke chaos. For a zero entropy interval map, it is shown that a pair is a sequence entropy pair if and only if it is f-nonseparable. Moreover, some equivalent conditions of positive entropy which relate to the number “3” are obtained. It is shown that for an interval map if it is topological null, then the pattern entropy of every open cover is of polynomial order, answering a question by Huang and Ye when the space is the closed unit interval.

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. Balibrea F., Lopez V.J.: A charactization of chaotic functions with entropy zero via their maximal scrambled sets. Math. Bohemica 120(3), 293–298 (1995)

    MATH  Google Scholar 

  2. Blanchard, F.: Fully Positive Topological Entropy and Topological Mixing, Symbolic Dynamics and its Applications, vol. 135 (ContemporaryMathematics), pp. 95–105. American Mathematical Society, Providence, RI (1992)

  3. Blanchard F., Host B., Maass A.: Topological complexity. Ergod. Theory Dynam. Sys. 20, 641–662 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blanchard F., Glasner E., Kolyada S., Maass A.: On Li–Yorke pairs. J. Reine Angew. Math. 547, 51–68 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Block A., Bruckner A.M., Humer P.D., Smital J.: The space of ω-limit sets of a continuous map of the interval. Trans. Am. Math. Soc. 384(4), 1357–1372 (1996)

    Google Scholar 

  6. Block L.S., Copple W.A.: Dynamics in One Dimemsion. Lecture Notes in Mathematics, vol. 1513. Springer, NY (1992)

    Google Scholar 

  7. Devaney R.L.: An Introduction to Chaotic Dynamical Systems. 2nd edn. Addison-Wesley Publishing Company Advanced Book Program, RedwoodCity, CA (1989)

    MATH  Google Scholar 

  8. Franzova N., Smital J.: Positive sequence topological entropy characterizes chaotic maps. Proc. Am. Math. 112(4), 1083–1086 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Glasner E.: On tame dynamical systems. Colloq. Math. 105, 283–295 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Glasner E., Ye X.: Local entropy theory. Ergod. Theory Dynam. Sys. 29(2), 321–356 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Huang W., Ye X.: Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topol. Appl. 117(3), 259–272 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Huang W., Ye X.: A local variational relation and applications. Israel J. Math. 151, 237–280 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Huang W., Ye X.: Combinatorial lemmas and applications to dynamics. Adv. Math. 220(6), 1689–1716 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jankova K., Smital J.: A characterization of chaos. Bull. Aust. Math. Soc. 34(2), 283–292 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Kuchta, M., Smital, J.: Two-point scrambled set implies chaos. In: European Conference on Iteration Theroy (Caldes De Malavella, 1987), pp. 427–430. World Sci. Publishing, Teaneck, NJ (1989)

  17. Li S.: ω-chaos and topological entropy. Trans. Am. Math. Soc. 339(1), 243–249 (1993)

    Article  MATH  Google Scholar 

  18. Li T., Yorke J.: Period three implies chaos. Am. Math. Monthly 82(10), 985–992 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  19. Schweizer B., Smital J.: Measure of chaos and a spectral decomposition of dynamical systems on the interval. Trans. Am. Math. Soc. 344, 737–754 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sharkovsky A.N.: The partially ordered system of attracting sets. Soviet Math. Dokl. 7(5), 1384–1386 (1966)

    Google Scholar 

  21. Ruette, S.: Chaos for Continuous Interval Maps: A Survey of Relationship Between Thevarious Sorts of Chaos (2003). http://www.math.u-psud.fr/~ruette/

  22. Shao S., Ye X., Zhang R.: Sensitivity and regionally proximal relation in minimal systems. Sci. China A 51, 987–994 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sharkovskii A.N.: On a theorem of G.Birkhoff. Dopovidi Akad. Nauk Ukrain RSR A 5, 423–429 (1967)

    Google Scholar 

  24. Smital J.: Chaotic functions with zero topological entropy. Trans. Am. Math. Soc. 297(1), 269–282 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tan F., Ye X., Zhang R.: The set of sequence entropy for a given space. Nonlinearity 23, 159–178 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Xiong J.: A chaotic map with topological entropy [zero]. Acta Math. Sci. (English Ed.) 6(4), 439–443 (1986)

    MathSciNet  MATH  Google Scholar 

  27. Xiong J.: Chaos in topological transitive systems. Sci. China A 48, 929–939 (2005)

    Article  MATH  Google Scholar 

  28. Xiong, J., Yang, Z.: Chaos Caused by a Toplogical Mixing Map, Dynamical Systems and Related Topics (Nagoya, 1990), Series in Dynamical Systems, vol. 9, pp. 550–572. World Scientific, River Edge, NJ (1991)

  29. Ye X., Zhang R.: On sensitive sets in topological dynamics. Nonlinearity 21(7), 1601–1620 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China

    Jian Li

Authors
  1. Jian Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian Li.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, J. Chaos and Entropy for Interval Maps. J Dyn Diff Equat 23, 333–352 (2011). https://doi.org/10.1007/s10884-011-9206-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-011-9206-5

Keywords

Mathematics Subject Classification (2000)

Search

Navigation