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Lectures on Lyapunov exponents and smooth ergodic theory. (With two appendices by M. Brin, D. Dolgopyat, H. Hu and Ya. Pesin). (English) Zbl 0996.37001

Katok, Anatole (ed.) et al., Smooth ergodic theory and its applications. Proceedings of the AMS summer research institute, Seattle, WA, USA, July 26-August 13, 1999. Providence, RI: American Mathematical Society (AMS). Proc. Symp. Pure Math. 69, 3-106 (2001).
This interesting survey gives a clear and detailed introduction to the theory of Lyapunov exponents and to some applications of the theory.
Contents: 1. Lyapunov exponents for differential equations. 2. Abstract theory of Lyapunov exponents. 3. Regularity of Lyapunov exponents associated with differential equations. 4. Lyapunov stability theory. 5. The Oseledets decomposition. 6. Dynamical systems with nonzero Lyapunov exponents. Multiplicative ergodic theorem. 7. Nonuniform hyperbolicity. Regular sets. 8. Examples of nonuniformly hyperbolic systems. 9. Existence of local stable manifolds. 10. Basic properties of local stable and unstable manifolds. 11. Absolute continuity. Holonomy map. 12. Absolute continuity and smooth invariant measures. 13. Ergodicity of nonuniformly hyperbolic systems preserving smooth measures. 14. Local ergodicity. 15. The entropy formula. 16. Ergodic properties of geodesic flows on compact surfaces of nonpositive curvature.
The survey contains two appendices: Hölder continuity of invariant distributions by M. Brin, and an example of a smooth hyperbolic measure with countably many ergodic components by D. Dolgopyat, H. Hu and Ya. Pesin.
For the entire collection see [Zbl 0973.00044].

MSC:

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37A30 Ergodic theorems, spectral theory, Markov operators
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
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