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On the Set of Continuity of the Topological Entropy of Parameter-Dependent Mappings of the Interval

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Abstract

Families of continuous mappings of the interval continuously depending on a parameter are considered. Any \(G_\delta\) set dense in the parameter space is realized as the set of continuity of topological entropy for a suitable family of continuous mappings.

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References

  1. R. L. Adler, A. G. Konheim, and M. H. McAndrew, “Topological entropy”, Trans. Amer. Math. Soc., 114 (1965), 309–319.

    Article  MathSciNet  Google Scholar 

  2. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge, 1995.

    Book  Google Scholar 

  3. A. N. Vetokhin, “Typical property of the topological entropy of continuous mappings of compact sets”, Differ. Uravn., 53:4 (2017), 448–453; English transl.:, Differ. Equations, 53:4 (2017), 439–444.

    MathSciNet  MATH  Google Scholar 

  4. A. N. Vetokhin, “The set of lower semi-continuity points of topological entropy of a continuous one-parametric family of dynamical systems”, Vestn. Mosk. Univ. Ser. 1. Mat., Mekh., :3 (2019), 69–72; English transl.:, Moscow Univ. Math. Bull., 74:3 (2019), 131–133.

    MathSciNet  MATH  Google Scholar 

  5. A. N. Vetokhin, “On some properties of topological entropy and topological pressure of families of dynamical systems continuously depending on a parameter”, Differ. Uravn., 55:10 (2019), 1319–1327; English transl.:, Differ. Equations, 55:10 (2019), 1275–1283.

    MathSciNet  MATH  Google Scholar 

  6. M. Misiurewicz, “Horseshoes for mappings of the interval”, Bull. Acad. Polon. Sci., Ser. Math. Astron. et Phys., 27:2 (1979), 167–169.

    MathSciNet  MATH  Google Scholar 

  7. F. Hausdorff, Mengenlehre, Walter de Gruyter, Berlin, Leipzig, 1927.

    MATH  Google Scholar 

  8. L. Block, “Noncontinuity of topological entropy of maps of the Cantor set and of the interval”, Proc. Amer. Math. Soc., 50 (1975), 388–393.

    Article  MathSciNet  Google Scholar 

  9. K. Kuratowski, Topologie, Polish Math. Soc., Warszawa, Lwów, 1948.

    MATH  Google Scholar 

  10. M. Misiurewicz and W. Szlenk, “Entropy of piecewise monotone mappings”, Studia Math., 67:1 (1980), 45–63.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    A. N. Vetokhin

  2. Bauman Moscow State Technical University, Moscow, Russia

    A. N. Vetokhin

Authors
  1. A. N. Vetokhin
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Correspondence to A. N. Vetokhin.

Additional information

Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2021, Vol. 55, pp. 42–50 https://doi.org/10.4213/faa3841.

Translated by O. V. Sipacheva

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Vetokhin, A.N. On the Set of Continuity of the Topological Entropy of Parameter-Dependent Mappings of the Interval. Funct Anal Its Appl 55, 210–216 (2021). https://doi.org/10.1134/S0016266321030035

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  • DOI: https://doi.org/10.1134/S0016266321030035

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