  In the stability theory of dynamical systems, Lyapunov functions play a
fundamental role. In this paper, we study the attractor-repeller pair
decomposition and Morse decomposition for compact metric space in the random
setting. In contrast to [8], by introducing slightly stronger definitions of
random attractor and repeller, we characterize attractor-repeller pair
decompositions and Morse decompositions for random dynamical systems through
the existence of Lyapunov functions. These characterizations, we think, deserve
to be known widely.

Dynamical Systems