Furstenberg's theorem for nonlinear stochastic systems

We extend Furstenberg's theorem to the case of an i.i.d. random composition of incompressible diffeomorphisms of a compact manifold M. The original theorem applies to linear maps {X_i}_i∈N on ℝ^m with determinant 1, and says that the highest Lyapunov exponent {Mathematical expression} is strictly positive unless there is a probability measure on the projective (m-1)-space which is a.s. invariant under the action of X_i. Our extension refers to a probability measure on the projective bundle over M. We show that when our diffeomorphism is the flow of a stochastic differential equation, the criterion for β>0 is ensured by a Lie algebra condition on the induced system on the principal bundle over M. © 1987 Springer-Verlag.