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Given a measurable dynamical system with an invariant measure, the measure of the points which return infinitely often to a set is not smaller than the measure of the set itself. This is the Poincar~ recurrence theorem. Given a continuous dynamical system, one can introduce the ~-limit set of a set in the state space. This is the set of all points which are approached arbitrarily close by orbits starting from the initial set. It can be considered as ,,the opposite, of the set of recurrent points. We show fLrSt that for an invariant measure for a flow of continuous maps on a topological state space the measure of the ~-limit set is not smaller than the measure of the set itself. Then we extend the result to random dynamical systems. This is then used to derive that for random dynamical systems on Polish spaces which have a (random) attractor every invariant measure is supported by the attractor. This already holds ff the attractor attracts only compact sets (and not necessarily bounded sets, which makes a difference for infinite-dimensional systems).We then address the question whether ~-limit sets of deterministic bounded sets (*) Entrata in Redazione il 10 marzo 1997. Indirizzo dell'A-:

The notion of an attractor for a random dynamical system with respect to a general collection of deterministic sets is introduced. This comprises, in particular, global point attractors and global set attractors. After deriving a necessary and sufficient condition for existence of the corresponding attractors it is proved that a global set attractor always contains all unstable sets of all of its subsets. Then it is shown that in general random point attractors, in contrast to deterministic point attractors, do not support all invariant measures of the system. However, for white noise systems it holds that the minimal point attractor supports all invariant Markov measures of the system.

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The theories of nonautonomous and random dynamical systems have undergone extensive, often parallel, developments in the past two decades. In particular, new concepts of nonautonomous and random attractors have been introduced. These consist of families of sets that are mapped onto each other as time evolves and have two forms: a forward attractor based on information about the system in the future and a pullback attractor that uses information about the past of the system. Both reduce to the usual attractor consisting of a single set in the autonomous case.Keywords Skew product flow · 2-parameter semi-group · Pullback attractor · Forward attractor · Random dynamical system · Weak attractor · Mean-square random dynamical system · Mean-field stochastic differential equations Mathematics Subject Classification (2010) 34D45 · 35B41 · 37-02 · 37B55 · 37C70 · 37H99 · 37L30 · 37L55 · 60H10 · 60H15