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Probabilistic ideas in smooth dynamical systems. (English) Zbl 0561.58025
Group theoretical methods in physics, 13th Int. Colloq., College Park/Md. 1984, 40-53 (1984).
[For the entire collection see Zbl 0547.00049.]
This survey paper discusses several theorems concerning smooth dynamical systems \((f^ n)\) on a compact manifold M. Emphasis is given in particular on Bowen-Ruelle-Sinai measures, i.e. f-invariant probability measures \(\mu\) on M such that, for all x in some positive Lebesgue measure subset, and all continuous \(\phi\) : \(M\to {\mathbb{R}}\), the ”time average” \(\lim_{n\to \infty}1/n\sum_{0\leq k<n}\phi (f^ kx)\) exists and equals \(\int \phi d\mu\). Relations of this notion with characteristic exponents, hyperbolic sets... are studied, without proofs.
Reviewer: F.Rouvi√®re

37A99 Ergodic theory
28D20 Entropy and other invariants