The Multiplicative Ergodic Theorem on Bundles and Manifolds

Summary

This chapter deals with extensions of the MET from matrix cocycles in ℝ^d to linear cocycles on bundles, in particular to the linearization of a nonlinear RDS on the tangent bundle, by which we prepare the smooth ergodic theory of Part III.

In order to facilitate the transfer of data of the MET from one bundle to another we need a “tempered” version of coordinate change called Lyapunov cohomology (Sect. 4.1).

In Sect. 4.2 the MET for the linearization of a nonlinear smooth RDS on a manifold is derived (Theorem 4.2.6). In case the RDS is generated by an RDE or SDE, we also give criteria in terms of the vector fields and the invariant measure ensuring the validity of the integrability conditions of the MET (Theorem 4.2.10 for the RDE case, Theorems 4.2.12, 4.2.13 and 4.2.14 for the SDE case).

Sect. 4.3 is devoted to one of the most important techniques of smooth ergodic theory, namely the use of (random) norms which “eat up” the non-uniformity of the MET (Theorem 4.3.6). It is of course crucial that the random norms do not, change the Lyapunov exponents (Corollary 4.3.10). We also obtain what is called the “strong version” of the MET (Theorem 4.3.12).