Preprint A384/2005
Almost all cocycles over any hyperbolic system have non-vanishing Lyapunov exponents

Marcelo Viana

**Keywords: **
Lyapunov exponent | uniform hyperbolicity | non-uniform hyperbolicity

We prove that for any $s>0$ the majority of $C^s$ linear cocycles
over any hyperbolic (uniformly or not) ergodic transformation
exhibit some non-zero Lyapunov exponent: this is true for an open
dense subset of cocycles and, actually, vanishing Lyapunov
exponents correspond to codimension-$\infty$. This open dense
subset is described in terms of a rather explicit geometric
condition involving the behavior of the cocycle over certain
homoclinic orbits of the transformation.