Lectures on Lyapunov exponents:
"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 si...
Gespeichert in:
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge, United Kingdom
Cambridge University Press
2014
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| Schriftenreihe: | Cambridge studies in advanced mathematics
145 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | "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".. |
| Beschreibung: | xiv, 202 Seiten Illustrationen |
| ISBN: | 9781107081734 |
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Datensatz im Suchindex
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| adam_text | Titel: Lectures on Lyapunov exponents
Autor: Viana, Marcelo
Jahr: 2014
Contents Preface page xi Introduction 1 1.1 Existence of Lyapunov exponents 1 1.2 Pinching and twisting 2 1.3 Continuity of Lyapunov exponents 3 1.4 Notes 3 1.5 Exercises 4 Linear cocycles 6 2.1 Examples 7 2.1.1 Products of random matrices 7 2.1.2 Derivative cocycles 8 2.1.3 Schrodinger cocycles 9 2.2 Hyperbolic cocycles 10 2.2.1 Definition and properties 10 2.2.2 Stability and continuity 14 2.2.3 Obstructions to hyperbolicity 16 2.3 Notes 18 2.4 Exercises 19 Extremal Lyapunov exponents 20 3.1 Subadditive ergodic theorem 20 3.1.1 Preparing the proof 21 3.1.2 Fundamental lemma 23 3.1.3 Estimating (p- 24 3.1.4 Bounding p + from above 26 3.2 Theorem of Furstenberg and Kesten 28 3.3 Herman’s formula 29 3.4 Theorem of Oseledets in dimension 2 30 Vll
30 34 36 36 38 38 40 40 41 44 47 48 52 53 53 55 56 59 59 61 63 64 67 67 70 75 77 79 81 85 85 86 89 91 91 96 97 102 102 Contents 3.4.1 One-sided theorem 3.4.2 Two-sided theorem 3.5 Notes 3.6 Exercises Multiplicative ergodic theorem 4.1 Statements 4.2 Proof of the one-sided theorem 4.2.1 Constructing the Oseledets flag 4.2.2 Measurability 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 4.4.2 Invariant cones 4.5 Notes 4.6 Exercises Stationary measures 5.1 Random transformations 5.2 Stationary measures 5.3 Ergodic stationary measures 5.4 Invertible random transformations 5.4.1 Lift of an invariant measure 5.4.2 .s-states and «-states 5.5 Disintegrations of .v-states and «-states 5.5.1 Conditional probabilities 5.5.2 Martingale construction 5.5.3 Remarks on 2-dimensional linear cocycles 5.6 Notes 5.7 Exercises Exponents and invariant measures 6.1 Representation of Lyapunov exponents 6.2 Furstenberg’s formula 6.2.1 Irreducible cocycles
Contents IX 6.2.2 Continuity of exponents for irreducible cocycles 103 6.3 Theorem of Furstenberg 105 6.3.1 Non-atomic measures 106 6.3.2 Convergence to a Dirac mass 108 6.3.3 Proof of Theorem 6.11 111 6.4 Notes 112 6.5 Exercises 113 7 Invariance principle 115 7.1 Statement and proof 116 7.2 Entropy is smaller than exponents 117 7.2.1 The volume case 118 7.2.2 Proof of Proposition 7.4. 119 7.3 Furstenberg’s criterion 124 7.4 Lyapunov exponents of typical cocycles 125 7.4.1 Eigenvalues and eigenspaces 126 7.4.2 Proof of Theorem 7.12 128 7.5 Notes 130 7.6 Exercises 131 8 Simplicity 133 8.1 Pinching and twisting 133 8.2 Proof of the simplicity criterion 134 8.3 Invariant section 137 8.3.1 Grassmannian structures 137 8.3.2 Linear arrangements and the twisting property 139 8.3.3 Control of eccentricity 140 8.3.4 Convergence of conditional probabilities 143 8.4 Notes 147 8.5 Exercises 147 9 Generic cocycles 150 9.1 Semi-continuity 151 9.2 Theorem of Mane-Bochi 153 9.2.1 Interchanging the Oseledets subspaces 155 9.2.2 Coboundary sets 157 9.2.3 Proof of Theorem 9.5 160 9.2.4 Derivative cocycles and higher dimensions 161 9.3 Holder examples of discontinuity 164 9.4 Notes 168 9.5 Exercises 169
X Contents Continuity 171 10.1 Invariant subspaces 172 10.2 Expanding points in projective space 174 10.3 Proof of the continuity theorem 176 10.4 Couplings and energy 178 10.5 Conclusion of the proof 181 10.5.1 Proof of Proposition 10.9 183 10.6 Final comments 186 10.7 Notes 189 10.8 Exercises 189 References 191 Index 198
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| illustrated | Illustrated |
| indexdate | 2024-07-10T01:11:12Z |
| institution | BVB |
| isbn | 9781107081734 |
| language | English |
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| physical | xiv, 202 Seiten Illustrationen |
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| series | Cambridge studies in advanced mathematics |
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| spelling | Viana, Marcelo 1962- Verfasser (DE-588)1073175278 aut Lectures on Lyapunov exponents Marcelo Viana (Instituto Nacional de Mathemática Pura e Aplicada - IMPA, Rio de Janeiro) Cambridge, United Kingdom Cambridge University Press 2014 xiv, 202 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 145 "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".. MATHEMATICS / Differential Equations bisacsh Lyapunov exponents MATHEMATICS / Differential Equations Ljapunov-Exponent (DE-588)4123668-3 gnd rswk-swf Ljapunov-Exponent (DE-588)4123668-3 s DE-604 Cambridge studies in advanced mathematics 145 (DE-604)BV000003678 145 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027482885&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Viana, Marcelo 1962- Lectures on Lyapunov exponents Cambridge studies in advanced mathematics MATHEMATICS / Differential Equations bisacsh Lyapunov exponents MATHEMATICS / Differential Equations Ljapunov-Exponent (DE-588)4123668-3 gnd |
| subject_GND | (DE-588)4123668-3 |
| title | Lectures on Lyapunov exponents |
| title_auth | Lectures on Lyapunov exponents |
| title_exact_search | Lectures on Lyapunov exponents |
| title_full | Lectures on Lyapunov exponents Marcelo Viana (Instituto Nacional de Mathemática Pura e Aplicada - IMPA, Rio de Janeiro) |
| title_fullStr | Lectures on Lyapunov exponents Marcelo Viana (Instituto Nacional de Mathemática Pura e Aplicada - IMPA, Rio de Janeiro) |
| title_full_unstemmed | Lectures on Lyapunov exponents Marcelo Viana (Instituto Nacional de Mathemática Pura e Aplicada - IMPA, Rio de Janeiro) |
| title_short | Lectures on Lyapunov exponents |
| title_sort | lectures on lyapunov exponents |
| topic | MATHEMATICS / Differential Equations bisacsh Lyapunov exponents MATHEMATICS / Differential Equations Ljapunov-Exponent (DE-588)4123668-3 gnd |
| topic_facet | MATHEMATICS / Differential Equations Lyapunov exponents Ljapunov-Exponent |
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