The random case of Conley's theorem: II. The complete Lyapunov function

Abstract

Conley (1978 Isolated Invariant Sets and the Morse Index (Conf. Board Math. Sci. vol 38) (Providence, RI: American Mathematical Society)) constructed a complete Lyapunov function for a flow on compact metric space which is constant on orbits in the chain recurrent set and strictly decreasing on orbits outside the chain recurrent set. This indicates that the dynamical complexity focuses on the chain recurrent set and the dynamical behaviour outside the chain recurrent set is quite simple. In this paper, a similar result is obtained for random dynamical systems (RDS) under the assumption that the base space is a separable metric space endowed with a probability measure. By constructing a complete Lyapunov function, which is constant on orbits in the random chain recurrent set and strictly decreasing on orbits outside it, the random case of Conley's fundamental theorem of dynamical systems is obtained. Furthermore, this result for RDS is generalized to noncompact state spaces.