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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)

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.

Notes

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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Copyright information

© 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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