A geometric path from zero Lyapunov exponents to rotation cocycles

Abstract

We consider cocycles of isometries on spaces of nonpositive 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.

To appear in Ergodic Theory and Dynamical Systems. The title was reverted to a previous one. Some modifications were made according to the referee's suggestions. We also included an alternative proof of Theorem A

References in corpus (3)

• Barycenters of measures transported by stochastic flows
• Sur les rapprochements par conjugaison en dimension 1 et classe C^1
• Almost reduction and perturbation of matrix cocycles

Cited by in corpus (2)

• Semigroups of isometries of the hyperbolic plane
• Hyperbolization of cocycles by isometries of the euclidean space