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