确定性系统的统计性质

Chapter 1 Introduction

1.1 Discrete Deterministic Systems—from Order to Chaos

1.2 Statistical Study of Chaos

Exercises

Chapter 2 Foundations of Measure Theory

2.1 Measures and Integration

2.2 Basic Integration Theory

2.3 Functions of Bounded Variation in One Variable

2.4 Functions of Bounded Variation in Several Variables

2.5 Compactness and Quasi—compactness

2.5.1 Strong and Wleak Compactness

2.5.2 Quasi-Compactness

Exercises

Chapter 3 Rudiments of Ergodic Theory

3.1 Measure Preserving TransfcIrmations

3.2 Ergodicity,Mixing and Exactness

3.2.1 Ergodicity

3.2.2 Mixing and Exactness

3.3 Ergodic Theorems

3.4 Topological Dynamical Systems

Exercises

Chapter 4 Frobenius-Perron Operators

4.1 Markov Operatorst

4.2 nobenius—Perron Operators

4.3 Koopman 0peratorst

4.4 Ergodicity and Frobenius—Perron Operators

4.5 Decomposition Theorem and Spectral Analysis

Exercises

Chapter 5 Invariant Measures——Existence

5.1 General Existence Results

5.2 Piecewise Stretching Mappings

5.3 Piecewise Convex Mappings

5.4 Piecewise Expanding Transformations

Exercises

Chapter 6 Invariant Measures--Computation

61 Ulam’s Method for One—Dimensional Mappings

6.2 Ulam’S Method for N—dimensional Transformations

6.3 The Markov Method for One—Dimensional Mappings

6.4 The Markov Metho（~for N—dimensional Transformations

Exercises

Chapter 7 Convergence Rate Analysis

7.1 Error Estimates for Ulam’S Method.

7.2 More Explicit Error Estimates

7.3 Error Estimates for the Markov Method

Exercises

Chapter 8 Entropy

8.1 Shannon Entropy

8.2 Kolmogorov Entropy

8.3 Topological Entropy

8.4 Boltzmann Entropy

8.5 Boltzmann Entropy and Frobenius—Perron Operators

Exercises

Chapter 9 Applications of Invariant Measures

9.1 Decay of Correlations

9.2 Random Number Generationi

9.3 Conformational Dynamics of Bio—molecules4：

9.4 DS—CDMA in Wireless Communications

Exercises

Bibliography

Index

本书介绍的是确定性离散动力系统统计性质的基本理论与计算方法. 首先介绍了遍历理论的一些经典结果; 然后着重研究了对应于混沌映射的绝对连续不变测度的存在性与计算问题, 这归结于相应的Frobenius-Perron算子的泛函分析与数值分析; 最后本书介绍了Shannon熵、Kolmogorov熵、拓扑熵以及Boltzmann熵, 并给出了不变测度的一些最新应用.本书可作为数学、计算科学及工程专业的研究生教材或参考书.