Nonuniform Hyperbolicity
Dynamics of Systems with Nonzero Lyapunov Exponents

A self-contained, comprehensive account of modern smooth ergodic theory, the mathematical foundation of deterministic chaos.

Luis Barreira (Author), Yakov Pesin (Author)

9780521832588, Cambridge University Press

Hardback, published 3 September 2007

528 pages
23.4 x 15.6 x 3.3 cm, 0.99 kg

'… will be indispensable for any mathematically inclined reader with a serious interest in the subject.' EMS Newsletter

Designed to work as a reference and as a supplement to an advanced course on dynamical systems, this book presents a self-contained and comprehensive account of modern smooth ergodic theory. Among other things, this provides a rigorous mathematical foundation for the phenomenon known as deterministic chaos - the appearance of 'chaotic' motions in pure deterministic dynamical systems. A sufficiently complete description of topological and ergodic properties of systems exhibiting deterministic chaos can be deduced from relatively weak requirements on their local behavior known as nonuniform hyperbolicity conditions. Nonuniform hyperbolicity theory is an important part of the general theory of dynamical systems. Its core is the study of dynamical systems with nonzero Lyapunov exponents both conservative and dissipative, in addition to cocycles and group actions. The results of this theory are widely used in geometry (e.g., geodesic flows and Teichmüller flows), in rigidity theory, in the study of some partial differential equations (e.g., the Schrödinger equation), in the theory of billiards, as well as in applications to physics, biology, engineering, and other fields.

Part I. Linear Theory: 1. The concept of nonuniform hyperbolicity
2. Lyapunov exponents for linear extensions
3. Regularity of cocycles
4. Methods for estimating exponents
5. The derivative cocycle
Part II. Examples and Foundations of the Nonlinear Theory: 6. Examples of systems with hyperbolic behavior
7. Stable manifold theory
8. Basic properties of stable and unstable manifolds
Part III. Ergodic Theory of Smooth and SRB Measures: 9. Smooth measures
10. Measure-theoretic entropy and Lyapunov exponents
11. Stable ergodicity and Lyapunov exponents
12. Geodesic flows
13. SRB measures
Part IV. General Hyperbolic Measures: 14. Hyperbolic measures: entropy and dimension
15. Hyperbolic measures: topological properties.

Subject Areas: Nonlinear science [PBWR], Topology [PBP], Calculus & mathematical analysis [PBK]