A geometric path from zero Lyapunov exponents to rotation cocycles

Details A geometric path from zero Lyapunov exponents to rotation cocycles A geometric path from zero Lyapunov exponents to rotation cocycles

2011-12-02

arxiv.org/abs/1112.0397v2

Mathematics

Dynamical Systems

37 pages, 4 figures. This version: some typos were corrected

Scientific paper

We consider cocycles of isometries on spaces of nonpositive curvature $H$. We show that if such a cocycle has uniform sublinear drift, then there are almost invariant sections, i.e., continuous sections that move very little under the cocycle dynamics. The construction is based on an appropriate concept of barycenter in $H$. If, in addition, $H$ is a symmetric space, then we show that almost invariant sections can be made invariant by perturbing the cocycle. Applying these results to the symmetric space $H = GL(d,\mathbb{R}) / O(d)$, we conclude that if all Lyapunov exponents of a matrix cocycle vanish, then it can be approximated by a cocycle that is cohomologous to a cocycle taking values in the orthogonal group $O(d)$. This extends a result of Avila, Bochi and Damanik to more general dynamics and arbitrary cocycle dimension. For linear cocycles with non-necessarily vanishing Lyapunov exponents, we show that the property of existence of conformal invariant structures along the Oseledets splitting is dense in the space of continuous cocycles.

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