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 on noncompact separable complete metric spaces, but under a so-called absorbing condition. In the present paper, the authors introduce a notion of random chain recurrent sets for random dynamical systems, and then prove the random Conley’s theorem on noncompact separable complete metric spaces without the absorbing condition.