# Preprint serie A 384/2005

Almost all cocycles over any hyperbolic system have non-vanishing Lyapunov exponents

Keywords:
Lyapunov exponent, uniform hyperbolicity, non-uniform hyperbolicity

Abstract:
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.

MSC 2000:
37C40    Smooth ergodic theory, invariant measures
37D25    Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37D30    Partially hyperbolic systems and dominated splittings
37H15    Multiplicative ergodic theory, Lyapunov exponents
37A35    Entropy and other invariants, isomorphism, classification
37C29    Homoclinic and heteroclinic orbits