Random Chain Recurrent Sets for Random Dynamical Systems

TL;DR

This paper extends Conley's theorem to random dynamical systems on noncompact spaces without requiring the absorbing condition, introducing a new notion of random chain recurrent sets.

Contribution

It introduces a novel concept of random chain recurrent sets and proves a version of Conley's theorem without the absorbing condition.

Findings

Established a new definition of random chain recurrent sets.

Proved a version of Conley's theorem for noncompact spaces without the absorbing condition.

Extended the theoretical framework of random dynamical systems.

Abstract

It is known by the Conley's theorem that the chain recurrent set $CR(\phi)$ of a deterministic flow $\phi$ 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 \emph{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 \emph{without} the absorbing condition.