In this chapter we introduce the crucial concept of regularity for cocycles over dynamical systems and we present one of our principal results – the Multiplicative Ergodic Theorem. It asserts that regularity of cocycles is a typical property in the measure-theoretic sense. We discuss various versions of this theorem as well as some of its manifestations including the Reduction Theorems. The latter describe normal forms of cocycles according to their Lyapunov exponents. The regularity of cocycles will also be used to deduce nonuniform hyperbolicity from absence of zero Lyapunov exponents.

The Lyapunov–Perron regularity

Let A be a cocycle over an invertible measurable transformation f of a measurable space X. Given x ϵ X, the cocycle A generates the sequence of matrices {Am}mϵℤ ={A(fm(x))}mϵℤ.

Definition 3.1.1. We say that x is a forward (respectively, backward) regular point for A if the sequence of matrices {A(fm(x))}mϵℤ is forward (respectively, backward) regular (see Section 1.3.2).

One can easily check that if x is a forward (respectively, backward) regular point for A then for every m ϵ ℤ, the point fm(x) is also forward (respectively, backward) regular for A. Furthermore, if A and ℬ are equivalent cocycles on Y then the point y ϵ Y is forward (respectively, backward) regular for A if and only if it is forward (respectively, backward) regular for ℬ.