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 $\bar{f}:K\left(X\right)⟶K\left(X\right)$, on the hyperspace of non-empty compact subsets of X, and $\hat{f}:F\left(X\right)⟶F\left(X\right)$, on the space of normal fuzzy sets consisting of the upper-semicontinuous functions $u:X⟶[0,1]$ with compact support and such that $u^{-1}\left(\right\{1\left\}\right)\ne \varnothing$. In particular, we characterize the properties of topological and multiple recurrence for the extended systems $\left(K\right(X),\bar{f})$ and $\left(F\right(X),\hat{f})$, 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.