Lectures on Lyapunov Exponents

In Chapters 7 and 8 we came to the conclusion that, for a significant class of linear cocycles, the Lyapunov exponents are most of the time distinct. The results were stated for locally constant cocycles over Bernoulli shifts but, as observed at the end of both chapters, the conclusions extend much beyond: roughly speaking, they remain valid for Hölder-continuous cocycles with invariant holonomies, assuming that the base dynamics is sufficiently “chaotic”.

Rather in contrast, in the early 1980s Mañé [88] announced that generic (that is, a residual subset of all) area-preserving C1 diffeomorphisms on any surface have λ± = 0 at almost every point, or else they are Anosov diffeomorphisms. Actually, as observed in Example 2.10, the second alternative is possible only if the surface is the torus T2. A complete proof of Mañé's claim was first given by Bochi [30], based on an unpublished draft by Mañé himself. This family of ideas is the subject of the present chapter.

In Section 9.1, we make a few useful observations about semi-continuity of Lyapunov exponents. Then, in Section 9.2, we state and prove a version of the Mañé–Bochi theorem for continuous linear cocycles (Theorem 9.5). This can be extended in several ways: to the original setting of diffeomorphisms (derivative cocycles); to higher dimensions; and to continuous time systems. Some of these are briefly discussed in Section 9.2.4.