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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]

We consider asymptotic properties of families of multifunctions generated by cocycles, families of measures as well as Markov operators associated with such a mapping. The results we apply to random dynamical systems show connection of obtained attractors for families of multifunctions with global set attractors and also connection of supports of attracting measures for Markov families with global point attractors. The use of topological limits instead of standard Hausdorff distance lets us to get attractors under some quite general assumptions only on a cocycle mapping but without any assumptions on so-called parameter space and without standard assumption on existence of a compact absorbing set. [ABSTRACT FROM AUTHOR]

We introduce the notion of asymptotic stability of sequences of multifunctions associated with discrete cocycles. Some sufficient conditions for existence of attracting sets are given. The use of the topological (Kuratowski's) limits, as less complicated as commonly used Hausdorff metric, let us to weaken many standard assumptions. We show that in considered case existence of attractor is a property of a cocycle mapping itself and does not depend on properties of a parameter nor a state space. The obtained results generalize earlier on iterated function systems and can be applied for non-autonomous as well as random dynamical systems. [ABSTRACT FROM AUTHOR]