Bulletin of the Brazilian Mathematical Society

, Volume 37, Issue 3, pp 307–349 | Cite as

Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for SL(2, ℝ) cocycles



We consider the linear cocycle (T, A) induced by a measure preserving dynamical system T : XX and a map A: XSL(2, ℝ). We address the dependence of the upper Lyapunov exponent of (T, A) on the dynamics T when the map A is kept fixed. We introduce explicit conditions on the cocycle that allow to perturb the dynamics, in the weak and uniform topologies, to make the exponent drop arbitrarily close to zero.

In the weak topology we deduce that if X is a compact connected manifold, then for a Cr (r ≥ 1) open and dense set of maps A, either (T, A) is uniformly hyperbolic for every T, or the Lyapunov exponents of (T, A) vanish for the generic measurable T.

For the continuous case, we obtain that if X is of dimension greater than 2, then for a Cr (r ≥ 1) generic map A, there is a residual set of volume-preserving homeomorphisms T for which either (T, A) is uniformly hyperbolic or the Lyapunov exponents of (T, A) vanish.


linear cocycles Lyapunov exponents uniform hyperbolicity volume-preserving homeomorphisms 

Mathematical subject classification:

Primary 37H15 Secondary 37A05 

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Instituto de Matemática—UFRGSPorto AlegreBRAZIL
  2. 2.LAGA UMR 7539, Université Paris13VilletaneuseFRANCE

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