Stability of Oseledets spaces is equivalent to stability of Lyapunov exponents

The objects of the multiplicative ergodic theorem (Lyapunov exponents and Oseledets spaces for products of random matrices) play a key role in the local theory of random dynamical systems. In this paper their behaviour under perturbations is investigated. It is shown that stability of Oseledets spaces is equivalent to stability of Lyapunov exponents.