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

** 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