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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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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