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.
Chapter
2.1.1 Products of random matrices
2.1.2 Derivative cocycles
2.1.3 Schrddotodinger cocycles
2.2.1 Definition and properties
2.2.2 Stability and continuity
2.2.3 Obstructions to hyperbolicity
3
Extremal Lyapunov exponents
3.1 Subadditive ergodic theorem
3.1.1 Preparing the proof
3.1.3 Estimating varphi[sub(–)]
3.1.4 Bounding varphi[sub(+)] from above
3.2 Theorem of Furstenberg and Kesten
3.4 Theorem of Oseledets in dimension 2
4
Multiplicative ergodic theorem
4.2 Proof of the one-sided theorem
4.2.1 Constructing the Oseledets flag
4.2.3 Time averages of skew products
4.2.4 Applications to linear cocycles
4.2.5 Dimension reduction
4.2.6 Completion of the proof
4.3 Proof of the two-sided theorem
4.3.1 Upgrading to a decomposition
4.3.2 Subexponential decay of angles
4.3.3 Consequences of subexponential decay
4.4 Two useful constructions
4.4.1 Inducing and Lyapunov exponents
5.1 Random transformations
5.3 Ergodic stationary measures
5.4 Invertible random transformations
5.4.1 Lift of an invariant measure
5.4.2 s-states and u-states
5.5 Disintegrations of s-states and u-states
5.5.1 Conditional probabilities
5.5.2 Martingale construction
5.5.3 Remarks on 2-dimensional linear cocycles
6
Exponents and invariant measures
6.1 Representation of Lyapunov exponents
6.2 Furstenberg’s formula
6.2.1 Irreducible cocycles
6.2.2 Continuity of exponents for irreducible cocycles
6.3 Theorem of Furstenberg
6.3.1 Non-atomic measures
6.3.2 Convergence to a Dirac mass
6.3.3 Proof of Theorem 6.11
7.2 Entropy is smaller than exponents
7.2.2 Proof of Proposition 7.4.
7.3 Furstenberg’s criterion
7.4 Lyapunov exponents of typical cocycles
7.4.1 Eigenvalues and eigenspaces
7.4.2 Proof of Theorem 7.12
8.1 Pinching and twisting
8.2 Proof of the simplicity criterion
8.3.1 Grassmannian structures
8.3.2 Linear arrangements and the twisting property
8.3.3 Control of eccentricity
8.3.4 Convergence of conditional probabilities
9.2 Theorem of Mantildenacutee–Bochi
9.2.1 Interchanging the Oseledets subspaces
9.2.3 Proof of Theorem 9.5
9.2.4 Derivative cocycles and higher dimensions
9.3 Hddotolder examples of discontinuity
10.2 Expanding points in projective space
10.3 Proof of the continuity theorem
10.4 Couplings and energy
10.5 Conclusion of the proof
10.5.1 Proof of Proposition 10.9
Lectures on Lyapunov Exponents
( Cambridge Studies in Advanced Mathematics )
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