On fuzzy entropy and topological entropy of fuzzy extensions of dynamical systems

The goal of this paper is to establish connections between two entropy concepts, namely the fuzzy entropy of a fuzzy set, which appears in fuzzy set theory, and the topological entropy of Zadeh's extensions of continuous maps defined on metric compact spaces, which is a topological dynamics concept. We develop several concepts that allow us to simultaneously address the two entropy concepts. Except for stating several basic properties, we are partially capable of specifying in which classes of fuzzy sets dynamically complex behavior can appear by using these instruments. Such instruments are necessary for a proper understanding of the behavior of fuzzy systems generated from crisp, discrete dynamical systems.