Furstenberg's theorem for nonlinear stochastic systems

Abstract

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 {Xi}i∈N on ℝm with determinant 1, and says that the highest Lyapunov exponent $$\beta \equiv \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}||X_n \circ ... \circ X_1 ||$$ is strictly positive unless there is a probability measure on the projective (m-1)-space which is a.s. invariant under the action of Xi. Our extension refers to a probability measure on the projective bundle over M.