  We explore the concept of metastability in random dynamical systems,
focussing on connections between random Perron-Frobenius operator cocycles and
escape rates of random maps, and on topological entropy of random shifts of
finite type. The Lyapunov spectrum of the random Perron-Frobenius cocycle and
the random adjacency matrix cocycle is used to decompose the random system into
two disjoint random systems with rigorous upper and lower bounds on (i) the
escape rate in the setting of random maps, and (ii) topological entropy in the
setting of random shifts of finite type, respectively.
