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Abstract We present a new sufficient condition for existence of semiattractors for set-valued semiflows of state multifunctions associated with general cocycle mappings. We assume that a given cocycle has at least one weakly pullback contracting fiber. In particular, one can apply the result to iterated function systems having no contracting mappings. We also introduce and analyze a fiber structure of semiattractors. [ABSTRACT FROM AUTHOR]

In the paper we deal with minimal closed subsets invariant with respect to set-valued semiflow. Such sets are known as supports of invariant or even ergodic measures of stochastic processes associated with such semiflows. Our motivation comes from some earlier and recent results connected with bounded noise processes, but we work in the framework of set-valued semiflows with lower semicontinuous members on general metric space rather than mostly studied by many authors continuous and compact-valued ones. Such semiflows appear naturally when nonautonomous/random dynamical systems are considered. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]