On turbulent, erratic and other dynamical properties of Zadeh's extensions

► We study the relations between the dynamics of (X, f) and their set (fuzzy)-extensions. ► We prove that if (X, f) is turbulent then their set (fuzzy)-extensions are turbulent. ► We prove that if (X, f) is erratic then their set (fuzzy)-extensions are erratic. ► We provide examples showing that the reverse implications are not necessarily true.