Sensitivity and transitivity of fuzzified dynamical systems

In this paper, it is proved that a set-valued dynamical system is sensitively dependent on initial conditions (resp., F-sensitive, multi-sensitive) if and only if its g-fuzzification is sensitively dependent on initial conditions (resp., F-sensitive, multi-sensitive), where F is a Fürstenberg family. As an application, it is shown that there exists a sensitive dynamical system whose g-fuzzification does not have such sensitive dependence for any g in a certain domain. Moreover, a sufficient condition is derived for ensuring that the g-fuzzification of every nontrivial dynamical system is not transitive.