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.

Anexos:

• almost.pdf