A MULTIPLICATIVE ERGODIC THEOREM FOR LIPSCHITZ-MAPS

If (Fn, n ≥ 0) is a stationary (ergodic) sequence of Lipschitz maps of a locally compact Polish space X into itself having a.s. negative Lyapunov exponent function, the composition process Fn⋯F1x converges in distribution to a stationary (ergodic) process in X (independent of x). For every x, the empirical distribution of a trajectory converges with probability one, and for every ε>0, almost every trajectory is eventually within ε of the support. We use the fact that the Lyapunov exponent of a process "run backwards" is the same as forwards. A set invariance condition is given for the case when (Fn) is a Markov chain. The result has applications to computer graphics and stability in control theory. © 1990.