Invariant functions for the Lyapunov exponents of random matrices

A new approach to the study of Lyapunov exponents of random matrices is presented. We prove that any family of nonnegative (d x d)-matrices has a continuous concave invariant functional on R-+(d). Under some standard assumptions on the matrices, this functional is strictly positive, and the coefficient corresponding to it is equal to the largest Lyapunov exponent. As a corollary we obtain asymptotics for the expected value of the logarithm of norms of matrix products and of their spectral radii. Another corollary gives new upper and lower bounds for the Lyapunov exponent, and an algorithm for computing it for families of nonnegative matrices. We consider possible extensions of our results to general nonnegative matrix families and present several applications and examples.