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On Dynamics of Triangular Maps of the Square with Zero Topological Entropy

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Abstract

It is known that, for interval maps, zero topological entropy is equivalent with bounded topological sequence entropy as well as with the non-existence of Li–Yorke scrambled triples. In this paper we answer the question how the situation changes when triangular maps of the unit square are concerned instead of interval maps.

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References

  1. Block, L., Coppel, W.A.: Dynamics in One Dimension. Lecture Notes in Mathematics, vol. 1513. Springer, Berlin (1992)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Cánovas, J.S.: Topological sequence entropy of interval maps. Nonlinearity 17, 49–56 (2004)

    Article  MathSciNet  Google Scholar 

  4. de Melo, W., van Strien, S.: One-Dimensional Dynamics. Springer, Berlin (2012). ISBN 3642780431, 9783642780431

    MATH  Google Scholar 

  5. Downarowicz, T.: Entropy in Dynamical System. Cambridge University Press, Cambridge (2011). ISBN 1139500872, 9781139500876

    Book  Google Scholar 

  6. Downarowicz, T.: Minimal subsystems of triangular maps of type $2^\infty $. Conclusion of the Sharkovsky classification program. Chaos Solitons Fractals 49, 61–71 (2013)

    Article  MathSciNet  Google Scholar 

  7. Downarowicz, T., Štefánková, M.: Embedding Toeplitz systems in triangular maps: the last but one problem of the Sharkovsky classification program. Chaos Solitons Fractals 45, 1566–1572 (2012)

    Article  MathSciNet  Google Scholar 

  8. Goodman, T.N.T.: Topological sequence entropy. Proc. Lond. Math. Soc. 29, 331–350 (1974)

    Article  MathSciNet  Google Scholar 

  9. Huang, W., Li, J., Ye, X.: Stable sets and mean Li–Yorke chaos in positive entropy systems. J. Funct. Anal. 266, 3377–3394 (2014)

    Article  MathSciNet  Google Scholar 

  10. Kočan, Z.: The problem of classification of triangular maps with zero topological entropy. Ann. Math. Sil. 13, 181–192 (1999)

    MathSciNet  MATH  Google Scholar 

  11. Kolyada, S., Snoha, L.: Topological entropy of nonautonomous dynamical systems. Random Comput. Dyn. 4(2–3), 205–233 (1996)

    MathSciNet  MATH  Google Scholar 

  12. Li, J.: Chaos and entropy for interval maps. J. Dyn. Differ. Equ. 23, 333–352 (2011)

    Article  MathSciNet  Google Scholar 

  13. Štefánková, M.: The Sharkovsky program of classification of triangular maps—a survey. Topol. Proc. 48, 135–150 (2016)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The research was supported by Grant SGS/18/2016 from the Silesian University in Opava. Support of this institution is gratefully acknowledged. The author thanks his supervisor Professor Marta Štefánková for valuable suggestions and comments.

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  1. Mathematical Institute in Opava, Silesian university, Opava, Czech Republic

    Vojtěch Pravec

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  1. Vojtěch Pravec
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Correspondence to Vojtěch Pravec.

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Pravec, V. On Dynamics of Triangular Maps of the Square with Zero Topological Entropy. Qual. Theory Dyn. Syst. 18, 761–768 (2019). https://doi.org/10.1007/s12346-018-00311-7

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  • DOI: https://doi.org/10.1007/s12346-018-00311-7

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