Semiattractors of set-valued semiflows

We consider smallest closed invariant subsets of the phase space with respect to arbitrary set-valued semiflows. Such sets, called semiattractors, attract all trajectories of singletons but not necessary compact or bounded subsets. In particular, there are systems admitting semiattractors which do not have global attractors as usually understood. We show some sufficient conditions on the existence of such a set. We also prove lower semicontinuous dependence of semiattractors for strict semiflows of l.s.c. multifunction and the existence of semigroups of Markov operators generated by such semiflows. We are motivated by asymptotic properties of semiflows of multifunctions connected with iterated function systems as well as random dynamical systems.