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Global random attractors are uniquely determined by attracting deterministic compact sets. (English) Zbl 0954.37027

This paper deals with random dynamical systems (continuous) and their global attractors. The author shows that for an invariant measure for a flow of continuous maps on a topological state space the measure of the \(\Omega\)-limit set is not smaller than the measure of the set itself. Extending this result to random dynamical systems (RDS) the author derives that for RDS on Polish spaces a random set which attracts every (deterministic) compact set has full measure with respect to every invariant probability measure for the RDS. The author proves that there exists even a compact set whose \(\Omega\)-limit sets almost surely give the whole attractors provided that the base flow is ergodic. Moreover the author addresses the question of uniqueness of a random attractor even in the case, when the base flow is not ergodic.

MSC:

37H20 Bifurcation theory for random and stochastic dynamical systems
37A25 Ergodicity, mixing, rates of mixing
37H10 Generation, random and stochastic difference and differential equations
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