VI. Ergodic theory of differentiable dynamical systems: General systems

The ergodic theory of Axiom A dynamical systems has been extended to vastly more general situations, starting with the work of Oseledec [17] and Pesin [18, 19]. This has led to many developments, which are too technical to be usefully presented here. But it is worth mentioning that these developments took place at a time when the ideas of chaos were becoming popular, and they had an important impact on the mathematical research. For instance, a conjecture by Frederickson et al. [20] on a relation between dimension and Lyapunov exponents led to a rigorous inequality by Ledrappier [21]. The notion of SRB measure was extended by Ledrappier and Young who proved its equivalence with other definitions in a splendid piece of mathematical work [22]. For details we refer the reader to the review paper by Eckmann and Ruelle [23]. The research in this general area is exemplary in two respects. On one hand it shows how ideas coming from physics can provide vital inspiration to mathematics. On the other hand it makes it clear that competent mathematical theorem-proving (in fact hard technical work) is also necessary, and cannot be replaced by any amount of numerical computer experimentation or physical hand-waving.