Luis Barreira

Nonuniform Hyperbolicity (eBook, PDF)

Dynamics of Systems with Nonzero Lyapunov Exponents

• Format: PDF

Produktbeschreibung

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 Teichmuller flows), in rigidity theory, in the study of some partial differential equations (e.g., the Schrodinger equation), in the theory of billiards, as well as in applications to physics, biology, engineering, and other fields.

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Produktdetails

• Verlag: Cambridge University Press
• Erscheinungstermin: 3. September 2007
• Englisch
• ISBN-13: 9781107109483
• Artikelnr.: 39262354

Autorenporträt

Luis Barreira is a Professor of Mathematics at Instituto Superior Técnico in Lisbon. He obtained his PhD from The Pennsylvania State University in 1996, under the guidance of Yakov Pesin, with whom he co-authored the book Lyapunov Exponents and Smooth Ergodic Theory. He has also written two surveys and more than forty research papers in dynamical systems.

Inhaltsangabe

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.