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Nonuniform Hyperbolicity

Nonuniform Hyperbolicity
Dynamics of Systems with Nonzero Lyapunov Exponents


Part of Encyclopedia of Mathematics and its Applications

  • Date Published: November 2007
  • availability: Available
  • format: Hardback
  • isbn: 9780521832588

£ 141.00

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About the Authors
  • 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.

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

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    Product details

    • Date Published: November 2007
    • format: Hardback
    • isbn: 9780521832588
    • length: 528 pages
    • dimensions: 234 x 156 x 33 mm
    • weight: 0.99kg
    • availability: 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.

  • Authors

    Luis Barreira, Instituto Superior Técnico, Lisboa
    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.

    Yakov Pesin, Pennsylvania State University
    Yakov Pesin is a Distinguished Professor of Mathematics at The Pennsylvania State University. He obtained his PhD from The Gorky State University in 1979. He is the author of three books, Dimension Theory in Dynamical Systems, Lectures on Partial Hyperbolicity and Stable Ergodicity, and, with Luis Barreira, Lyapunov Exponents and Smooth Ergodic Theory, as well as six surveys and more than seventy research papers. He is an executive editor of the journal Ergodic Theory and Dynamical Systems.

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