On Topological Entropy of Zadeh’s Extension Defined on Piecewise Convex Fuzzy Sets

  • Jose Cánovas
  • Jiří KupkaEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 641)


As the main result of this article we prove that a given continuous interval map and its Zadeh’s extension (fuzzification) to the space of fuzzy sets with the property that \(\alpha \)-cuts have at most m convex (topologically connected) components, for m being an arbitrary natural number, have both positive (resp. zero) topological entropy. Presented topics are studied also for set-valued (induced) discrete dynamical systems. The main results are proved due to variational principle describing relations between topological and measure-theoretical entropy, respectively.



The first author has been supported by the grant MTM2014-52920-P from Ministerio de Economía y Competitividad (Spain). J. Kupka was supported by the NPU II project LQ1602 IT4Innovations excellence in science.


  1. 1.
    Acosta, G., Illanes, A., Méndez-Lango, H.: The transitivity of induced maps. Topology Appl. 159, 1013–1033 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aubin, D., Dahan Dalmedico, A.: Writing the history of dynamical systems and chaos: longue durée and revolution, disciplines and cultures. Historia Math. 29, 273–339 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Adler, R.L., Konheim, A.G., McAndrew, M.H.: Topological entropy. Trans. Am. Math. Soc. 114, 309–319 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blokh, A.: On the connection between entropy and transitivity for one-dimensional mappings. Russ. Math. Surv. 42, 209–210 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bowen, R.: Entropy for group endomorphism and homogeneous spaces. Trans. Am. Math. Soc. 153, 401–414 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cánovas, J.S., Kupka, J.: Topological entropy of fuzzified dynamical systems. Fuzzy Sets Syst. 165, 67–79 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cánovas, J.S., Kupka, J.: On the topological entropy on the space of fuzzy numbers. Fuzzy Sets Syst. 257, 132–145 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cánovas, J.S., Kupka, J.: On fuzzy entropy and topological entropy of fuzzy extensions of dynamical systems. Fuzzy Sets Syst. 309, 115–130 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cánovas, J., Rodríguez, J.M.: Topological entropy of maps on the real line. Topology Appl. 153, 735–746 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Denker, M., Grillenberger, C., Sigmund, K.: Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics, vol. 527. Springer-Verlag, New York (1976)Google Scholar
  11. 11.
    Diamond, P., Pokrovskii, A.: Chaos, entropy and a generalized extension principle. Fuzzy Sets Syst. 61, 277–283 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dumitrescu, D.: Fuzzy measures and the entropy of fuzzy partitions. J. Math. Anal. Appl. 176, 359–373 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dumitrescu, D.: Entropy of fuzzy dynamical systems. Fuzzy Sets Syst. 70, 45–57 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kolyada, S.: On dynamics of triangular maps of the square. Erg. Theory Dynam. Syst. 12, 749–768 (1992)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kupka, J.: On fuzzifications of discrete dynamical systems. Inf. Sci. 181(13), 2858–2872 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kupka, J.: On Devaney chaotic induced fuzzy and set-valued dynamical systems. Fuzzy Sets Syst. 177, 34–44 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Knopfmacher, J.: On measures of fuzziness. J. Math. Anal. Appl. 49, 529–534 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kwietniak, D., Oprocha, P.: Topological entropy and chaos for maps induced on hyperspaces. Chaos, Solitons Fractals 33, 76–86 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Román-Flores, H., Chalco-Cano, Y.: Some chaotic properties of Zadeh’s extension. Chaos, Solitons Fractals 35, 452–459 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Román-Flores, H., Chalco-Cano, Y., Silva, G.N., Kupka, G.J.: On turbulent, erratic and other dynamical properties of Zadehs extensions. Chaos, Solitons Fractals 44, 990–994 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics and StatisticsTechnical University of CartagenaCartagenaSpain
  2. 2.Institute for Reasearch and Applications of Fuzzy Modeling, NSC IT4InnovationsUniversity of OstravaOstravaCzech Republic

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