The random case of Conley’s theorem

The well-known Conley’s theorem states that the complement of chain recurrent set equals the union of all connecting orbits of the flow φ on the compact metric space X , i.e. X − CR(φ) = ⋃ [B(A) − A], where CR(φ) denotes the chain recurrent set of φ, A stands for an attractor and B(A) is the basin determined by A. In this paper we show that by appropriately selecting the definition of random attractor, in fact we define a random local attractor to be the ω-limit set of some random pre-attractor surrounding it, and by considering appropriate measurability, in fact we also consider the universal σ-algebra Fmeasurability besides F -measurability, we are able to obtain the random case of Conley’s theorem.