Statistical Properties of Deterministic Systems
by Ding, Jiu; Zhou, AihuiFormat: Hardcover
Pub. Date: 2009-07-04
Publisher(s): Springer Verlag
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Summary
Part of Tsinghua University Texts, "Statistical Properties of Deterministic Systems" discusses the fundamental theory and computational methods of the statistical properties of deterministic discrete dynamical systems. After introducing some basic results from ergodic theory, two problems related to the dynamical system are studied: first the existence of absolute continuous invariant measures, and then their computation. They correspond to the functional analysis and numerical analysis of the Frobenius-Perron operator associated with the dynamical system. The book can be used as a text for graduate students in applied mathematics and in computational mathematics; it can also serve as a reference book for researchers in the physical sciences, life sciences, and engineering. Dr. Jiu Ding is a professor at the Department of Mathematics of the University of Southern Mississippi; Dr. Aihui Zhou is a professor at the Academy of Mathematics and Systems Science of the Chinese Academy of Sciences.
Author Biography
Dr. Jiu Ding is a professor at the Department of Mathematics of the University of Southern Mississippi; Dr. Aihui Zhou is a professor at the Academy of Mathematics and Systems Science of the Chinese Academy of Sciences.
Table of Contents
| Introduction | p. 1 |
| Discrete Deterministic Systems-from Order to Chaos | p. 2 |
| Statistical Study of Chaos | p. 8 |
| Exercises | p. 14 |
| Foundations of Measure Theory | p. 15 |
| Measures and Integration | p. 15 |
| Basic Integration Theory | p. 21 |
| Functions of Bounded Variation in One Variable | p. 24 |
| Functions of Bounded Variation in Several Variables | p. 27 |
| Compactness and Quasi-compactness | p. 31 |
| Strong and Weak Compactness | p. 32 |
| Quasi-Compactness | p. 33 |
| Exercises | p. 35 |
| Rudiments of Ergodic Theory | p. 36 |
| Measure Preserving Transformations | p. 36 |
| Ergodicity, Mixing and Exactness | p. 39 |
| Ergodicity | p. 39 |
| Mixing and Exactness | p. 41 |
| Ergodic Theorems | p. 43 |
| Topological Dynamical Systems | p. 47 |
| Exercises | p. 59 |
| Frobenius-Perron Operators | p. 62 |
| Markov Operators | p. 63 |
| Frobenius-Perron Operators | p. 68 |
| Koopman Operators | p. 77 |
| Ergodicity and Frobenius-Perron Operators | p. 79 |
| Decomposition Theorem and Spectral Analysis | p. 84 |
| Exercises | p. 88 |
| Invariant Measures-Existence | p. 92 |
| General Existence Results | p. 92 |
| Piecewise Stretching Mappings | p. 99 |
| Piecewise Convex Mappings | p. 103 |
| Piecewise Expanding Transformations | p. 106 |
| Exercises | p. 113 |
| Invariant Measures-Computation | p. 115 |
| Ulam's Method for One-Dimensional Mappings | p. 116 |
| Ulam's Method for N-dimensional Transformations | p. 123 |
| The Markov Method for One-Dimensional Mappings | p. 127 |
| The Markov Method for N-dimensional Transformations | p. 134 |
| Exercises | p. 141 |
| Convergence Rate Analysis | p. 144 |
| Error Estimates for Ulam's Method | p. 144 |
| More Explicit Error Estimates | p. 152 |
| Error Estimates for the Markov Method | p. 161 |
| Exercises | p. 170 |
| Entropy | p. 172 |
| Shannon Entropy | p. 172 |
| Kolmogorov Entropy | p. 177 |
| Topological Entropy | p. 183 |
| Boltzmann Entropy | p. 186 |
| Boltzmann Entropy and Frobenius-Perron Operators | p. 189 |
| Exercises | p. 193 |
| Applications of Invariant Measures | p. 196 |
| Decay of Correlations | p. 196 |
| Random Number Generation | p. 199 |
| Conformational Dynamics of Bio-molecules | p. 204 |
| DS-CDMA in Wireless Communications | p. 215 |
| Exercises | p. 219 |
| Bibliography | p. 221 |
| Index | p. 228 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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