Rigorous numerical estimation of Lyapunov exponents and invariant measuresof iterated function systems and random matrix products

We present a fast, simple matrix method of computing the unique invariant m easure and associated Lyapunov exponents of a nonlinear iterated function s ystem. Analytic bounds for the error in our approximate invariant measure ( in terms of the Hutchinson metric) are provided, while convergence of the L yapunov exponent estimates to the true value is assured. As a special case, we are able to rigorously estimate the Lyapunov exponents of an lid random matrix product. Computation of the Lyapunov exponents is carried out by ev aluating an integral with respect to the unique invariant measure, rather t han following a random orbit. For low-dimensional systems, our method is co nsiderably quicker and more accurate than conventional methods of exponent computation. An application to Markov random matrix product is also describ ed.