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 f̄:K(X)(X), on the hyperspace of non-empty compact subsets of X, and f̂:F(X)(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\)≠. In particular, we characterize the properties of topological and multiple recurrence for the extended systems (K(X),f̄) and (F(X),f̂). Special attention is given to the case where the underlying space is completely metrizable, for which we obtain some stronger point-recurrence equivalences.