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Random chain recurrent sets for random dynamical systems

It is known by the Conley's theorem that the chain recurrent set CR(ϕ) of a deterministic flow ϕ on a compact metric space is the complement of the union of sets B(A) − A, where A varies over the collection of attractors and B(A) is the basin of attraction of A. It has recently been shown that a similar decomposition result holds for random dynamical systems (RDSs) on non-compact separable complete metric spaces, but under a so-called absorbing condition. In the present article, the authors introduce a notion of random chain recurrent sets for RDSs, and then prove the random Conley's theorem on non-compact separable complete metric spaces without the absorbing condition.