Robustness of exponential dichotomies in mean

Abstract

We consider the notion of an exponential dichotomy in mean, in which the exponential behavior in the classical notion of an exponential dichotomy is replaced by the much weaker requirement that the same happens in mean with respect to some probability measure. This includes as a special case any linear cocycle over a measure-preserving flow with nonzero Lyapunov exponents almost everywhere, such as the geodesic flow on a compact manifold of negative curvature. Our main aim is to show that the exponential behavior in mean is robust, in the sense that it persists under sufficiently small linear perturbations.