Comparison of Lyapunov Exponents and Boundaries

Abstract

In the preceding chapter we have given a criterion ensuring that the two upper Lyapunov exponents are distinct. It will give us all we need for the study of limit theorems. But a sharp study of the behaviour at infinity of the random products S requires a precise knowledge of the relations between all the exponents. For instance they provide all the limit values of $\frac{1}{n}\;Log\;\left\| {{S_{n}}\left( \omega \right)x} \right\|$ , when ω is kept fixed and x runs through ℝ^d (Osseledec’s theorem) and determine the possible boundaries towards which S_n converges.