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A geometric path from zero Lyapunov exponents to rotation cocycles

Published online by Cambridge University Press:  20 August 2013

JAIRO BOCHI  and
ANDRÉS NAVAS
JAIRO BOCHI
Affiliation:
PUC–Rio, Rua Marquês de S. Vicente, 225 Rio de Janeiro, Brazil email jairo@mat.puc-rio.br
ANDRÉS NAVAS
Affiliation:
Universidad de Santiago, Alameda 3363, Estación Central, Santiago, Chile email andres.navas@usach.cl
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Abstract

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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 2 , April 2015 , pp. 374 - 402
Copyright
© Cambridge University Press, 2013 

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