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

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Advances in Fuzzy Logic and Technology 2017 (EUSFLAT 2017, IWIFSGN 2017)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 641))

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Abstract

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.

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Acknowledgements

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.

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Technical University of Cartagena, C/ Doctor Flemming sn, 30202, Cartagena, Spain

    Jose Cánovas

  2. Institute for Reasearch and Applications of Fuzzy Modeling, NSC IT4Innovations, University of Ostrava, 30. dubna 22, Ostrava, Czech Republic

    Jiří Kupka

Authors
  1. Jose Cánovas
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  2. Jiří Kupka
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Corresponding author

Correspondence to Jiří Kupka .

Editor information

Editors and Affiliations

  1. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Janusz Kacprzyk

  2. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Eulalia Szmidt

  3. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Sławomir Zadrożny

  4. Department of Bioinformatics and Mathematical Modelling, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Krassimir T. Atanassov

  5. WIT - Warsaw School of Information Technology, Warsaw, Poland

    Maciej Krawczak

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Cánovas, J., Kupka, J. (2018). On Topological Entropy of Zadeh’s Extension Defined on Piecewise Convex Fuzzy Sets. In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K., Krawczak, M. (eds) Advances in Fuzzy Logic and Technology 2017. EUSFLAT IWIFSGN 2017 2017. Advances in Intelligent Systems and Computing, vol 641. Springer, Cham. https://doi.org/10.1007/978-3-319-66830-7_31

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  • DOI: https://doi.org/10.1007/978-3-319-66830-7_31

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66829-1

  • Online ISBN: 978-3-319-66830-7

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