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 $(\mathcal{ F}(X),\hat{ f})$, 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.
Categories: math.DS