Cocycles and Lyapunov Exponents

In this chapter, for cocycles over a measure-preserving transformation, we give a simple proof of the multiplicative ergodic theorem. The argument is based on the results on singular values established in Chap. 6 combined with the subadditive ergodic theorem. We also show how a nonvanishing Lyapunov exponent for a cocycle gives rise to nonuniform hyperbolicity. In particular, the structure that the theorem determines is fundamental in many developments of smooth ergodic theory. Finally, we show that some simpler parts of the theory can be extended to continuous maps, imitating results for the derivative cocycle of a differentiable map. Namely, we introduce numbers that play the role of the values of the Lyapunov exponent for maps that are not necessarily differentiable. These turn out to coincide on a repeller for a differentiable map, for almost every point with respect to an invariant measure.