Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents

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 nonzero 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 geometric condition involving the behavior of the cocycle over certain heteroclinic orbits of the transformation.