Zadeh's extension;fuzzy dynamical systems;topological entropy

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.