  We study for a dynamical system $f:X\longrightarrow X$ some of the principal
topological recurrence-kind properties with respect to the induced maps
$\overline{f}:\mathcal{K}(X)\longrightarrow\mathcal{K}(X)$, on the hyperspace
of non-empty compact subsets of $X$, and
$\hat{f}:\mathcal{F}(X)\longrightarrow\mathcal{F}(X)$, on the space of normal
fuzzy sets consisting of the upper-semicontinuous functions $u:X\longrightarrow
[0,1]$ with compact support and such that $u^{-1}(\{1\})\neq\varnothing$. In
particular, we characterize the properties of topological and multiple
recurrence for the extended systems $(\mathcal{K}(X),\overline{f})$ and
$(\mathcal{F}(X),\hat{f})$. Special attention is given to the case where the
underlying space is completely metrizable, for which we obtain some stronger
point-recurrence equivalences.
