The random case of Conley's theorem

Abstract

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. $X-\mathcal{CR}(ϕ)=\bigcup [B(A)-A]$, where $\mathcal{CR}(ϕ)$ 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 $\mathcal F^u$-measurability besides $\mathcal F$-measurability, we are able to obtain the random case of Conley's theorem.