Transitivity of some uniformities on fuzzy sets

Given a uniform space $(X,\mathcal{U})$, we denote by $\mathcal{F}(X)$ to the family of all normal upper semicontinuous fuzzy sets $u\text{\hspace{0.17em}⁣}:X\to [0,1]$ with compact support. In this paper, we study transitivity on some uniformities on $\mathcal{F}(X)$: the level-wise uniformity ${\mathcal{U}}_{\mathrm{\infty }}$, the Skorokhod uniformity ${\mathcal{U}}_0$, and the sendograph uniformity ${\mathcal{U}}_S$. If $f\text{\hspace{0.17em}⁣}:(X,\mathcal{U})\to (X,\mathcal{U})$ is a continuous function, we mainly characterize when the induced dynamical systems $\widehat{f}\text{\hspace{0.17em}⁣}:(\mathcal{F}(X),{\mathcal{U}}_{\mathrm{\infty }})\to (\mathcal{F}(X),{\mathcal{U}}_{\mathrm{\infty }})$, $\widehat{f}\text{\hspace{0.17em}⁣}:(\mathcal{F}(X),{\mathcal{U}}_0)\to (\mathcal{F}(X),{\mathcal{U}}_0)$ and $\widehat{f}\text{\hspace{0.17em}⁣}:(\mathcal{F}(X),{\mathcal{U}}_S)\to (\mathcal{F}(X),{\mathcal{U}}_S)$ are transitive, where $\widehat{f}$ is the Zadeh’s extension of $f$.