Furstenberg's theorem for nonlinear stochastic systems (Q1075694)

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 \(\{X_{i,i+1}\}_{i\in {\mathbb{N}}}\) on \({\mathbb{R}}^ m\) with determinant 1, and says that the highest Lyapunov exponent \[ \beta \equiv \lim_{n\to \infty}n^{-1}\| X_{n-1,n}\circ...\circ X_{0,1}\| \] is strictly positive unless there is a probability measure on the projective (m-1)-space which is a.s. invariant under the action of \(X_{i,i+1}\). 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 \(\beta >0\) is ensured by a Lie algebra condition on the induced system on the principal bundle over M.