Recurrence in collective dynamics: From the hyperspace to fuzzy dynamical systems

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