A Geometric Path from Zero Lyapunov Exponents to Rotation Cocycles

We consider cocycles of isometries on spaces of non-positive curvature H. We show that the supremum of the drift over all invariant ergodic probability measures equals the infimum of the displacements of continuous sections under the cocycle dynamics. In particular, if a cocycle has uniform sublinear drift, then there are almost invariant sections, that is, sections that move arbitrarily little under the cocycle dynamics. If, in addition, H is a symmetric space, then we show that almost invariant sections can be made invariant by perturbing the cocycle. © Cambridge University Press, 2013.