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Lectures on Lyapunov Exponents

Book Description

The theory of Lyapunov exponents originated over a century ago in the study of the stability of solutions of differential equations. Written by one of the subject's leading authorities, this book is both an account of the classical theory, from a modern view, and an introduction to the significant developments relating the subject to dynamical systems, ergodic theory, mathematical physics and probability. It is based on the author's own graduate course and is reasonably self-contained with an extensive set of exercises provided at the end of each chapter. This book makes a welcome addition to the literature, serving as a graduate text and a valuable reference for researchers in the field.

Table of Contents

  1. Cover Page
  2. Half Title Page
  3. Series Page
  4. Title Page
  5. Copyright
  6. Dedication
  7. Contents
  8. Preface
  9. 1 Introduction
    1. 1.1 Existence of Lyapunov exponents
    2. 1.2 Pinching and twisting
    3. 1.3 Continuity of Lyapunov exponents
    4. 1.4 Notes
    5. 1.5 Exercises
  10. 2 Linear cocycles
    1. 2.1 Examples
      1. 2.1.1 Products of random matrices
      2. 2.1.2 Derivative cocycles
      3. 2.1.3 Schrödinger cocycles
    2. 2.2 Hyperbolic cocycles
      1. 2.2.1 Definition and properties
      2. 2.2.2 Stability and continuity
      3. 2.2.3 Obstructions to hyperbolicity
    3. 2.3 Notes
    4. 2.4 Exercises
  11. 3 Extremal Lyapunov exponents
    1. 3.1 Subadditive ergodic theorem
      1. 3.1.1 Preparing the proof
      2. 3.1.2 Fundamental lemma
      3. 3.1.3 Estimating φ[sub(−)]
      4. 3.1.4 Bounding φ[sub(+)] from above
    2. 3.2 Theorem of Furstenberg and Kesten
    3. 3.3 Herman’s formula
    4. 3.4 Theorem of Oseledets in dimension 2
      1. 3.4.1 One-sided theorem
      2. 3.4.2 Two-sided theorem
    5. 3.5 Notes
    6. 3.6 Exercises
  12. 4 Multiplicative ergodic theorem
    1. 4.1 Statements
    2. 4.2 Proof of the one-sided theorem
      1. 4.2.1 Constructing the Oseledets flag
      2. 4.2.2 Measurability
      3. 4.2.3 Time averages of skew products
      4. 4.2.4 Applications to linear cocycles
      5. 4.2.5 Dimension reduction
      6. 4.2.6 Completion of the proof
    3. 4.3 Proof of the two-sided theorem
      1. 4.3.1 Upgrading to a decomposition
      2. 4.3.2 Subexponential decay of angles
      3. 4.3.3 Consequences of subexponential decay
    4. 4.4 Two useful constructions
      1. 4.4.1 Inducing and Lyapunov exponents
      2. 4.4.2 Invariant cones
    5. 4.5 Notes
    6. 4.6 Exercises
  13. 5 Stationary measures
    1. 5.1 Random transformations
    2. 5.2 Stationary measures
    3. 5.3 Ergodic stationary measures
    4. 5.4 Invertible random transformations
      1. 5.4.1 Lift of an invariant measure
      2. 5.4.2 s-states and u-states
    5. 5.5 Disintegrations of s-states and u-states
      1. 5.5.1 Conditional probabilities
      2. 5.5.2 Martingale construction
      3. 5.5.3 Remarks on 2-dimensional linear cocycles
    6. 5.6 Notes
    7. 5.7 Exercises
  14. 6 Exponents and invariant measures
    1. 6.1 Representation of Lyapunov exponents
    2. 6.2 Furstenberg’s formula
      1. 6.2.1 Irreducible cocycles
      2. 6.2.2 Continuity of exponents for irreducible cocycles
    3. 6.3 Theorem of Furstenberg
      1. 6.3.1 Non-atomic measures
      2. 6.3.2 Convergence to a Dirac mass
      3. 6.3.3 Proof of Theorem 6.11
    4. 6.4 Notes
    5. 6.5 Exercises
  15. 7 Invariance principle
    1. 7.1 Statement and proof
    2. 7.2 Entropy is smaller than exponents
      1. 7.2.1 The volume case
      2. 7.2.2 Proof of Proposition 7.4.
    3. 7.3 Furstenberg’s criterion
    4. 7.4 Lyapunov exponents of typical cocycles
      1. 7.4.1 Eigenvalues and eigenspaces
      2. 7.4.2 Proof of Theorem 7.12
    5. 7.5 Notes
    6. 7.6 Exercises
  16. 8 Simplicity
    1. 8.1 Pinching and twisting
    2. 8.2 Proof of the simplicity criterion
    3. 8.3 Invariant section
      1. 8.3.1 Grassmannian structures
      2. 8.3.2 Linear arrangements and the twisting property
      3. 8.3.3 Control of eccentricity
      4. 8.3.4 Convergence of conditional probabilities
    4. 8.4 Notes
    5. 8.5 Exercises
  17. 9 Generic cocycles
    1. 9.1 Semi-continuity
    2. 9.2 Theorem of Mañé–Bochi
      1. 9.2.1 Interchanging the Oseledets subspaces
      2. 9.2.2 Coboundary sets
      3. 9.2.3 Proof of Theorem 9.5
      4. 9.2.4 Derivative cocycles and higher dimensions
    3. 9.3 Hölder examples of discontinuity
    4. 9.4 Notes
    5. 9.5 Exercises
  18. 10 Continuity
    1. 10.1 Invariant subspaces
    2. 10.2 Expanding points in projective space
    3. 10.3 Proof of the continuity theorem
    4. 10.4 Couplings and energy
    5. 10.5 Conclusion of the proof
      1. 10.5.1 Proof of Proposition 10.9
    6. 10.6 Final comments
    7. 10.7 Notes
    8. 10.8 Exercises
  19. References
  20. Index