A geometric path from zero Lyapunov exponents to rotation cocycles

TL;DR

This paper explores the relationship between Lyapunov exponents and rotation cocycles in nonpositive curvature spaces, establishing conditions for the existence of almost invariant sections and their perturbations.

Contribution

It introduces a geometric framework connecting zero Lyapunov exponents to rotation cocycles, providing new insights into invariant sections in nonpositive curvature spaces.

Findings

Supremum of drift equals infimum of displacements for invariant measures

Existence of almost invariant sections under uniform sublinear drift

Almost invariant sections can be made invariant via perturbations in symmetric spaces

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.