Nonuniform Hyperbolicity
Nonuniform Hyperbolicity
$180.00
USDDesigned 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.
- The book summarizes and unifies results of smooth ergodic theory, which is one of the core parts of the general dynamical system theory
- Describes the theory of deterministic chaos
- The book can be used as supporting material for an advanced course on dynamical systems
Reviews & endorsements
'… will be indispensable for any mathematically inclined reader with a serious interest in the subject.' EMS Newsletter
Product details
September 2007Hardback
9780521832588
528 pages
234 × 156 × 33 mm
0.99kg
Available
Table of Contents
- 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.
