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ChaPter1 The foundation of topological dynamics1

第1章 拓扑动力系统基础1

§1 Topological dynamics1

§1．1 Dynamical Systems and Subsystems1

§1 动力系统和子系统1

§1．1 动力系统和子系统1

§1．2 映射空间5

§2．1 Recurrence5

§2 回复性5

§2．1 回复性5

§2 Recurrence6

§1．2 Mapping Spaces with the C0-topology6

§2．2 ω-limit sets8

§2．2 ω-极限集8

§3 拓扑传递性和拓扑混合性12

§3 Topological Transitivity and Mixing12

§3．1 Topological Transitivity12

§3．1 拓扑传递性12

§3．2 Topological Mixing14

§3．2 拓扑混合性14

§4．1 Minimal Sets and minimal Mappings15

§4 Minimal Sets and Almost Periodic Points15

§4 极小集和几乎周期点16

§4．2 Almost Periodic Points16

§4．2 几乎周期点16

§4．1 极小集和极小映射16

§5．1 Equivalent Classification of Systems——Topological Conjugacy18

§5 Topological conjugacy and semi-Conjugacy18

§5．1 紧致系统的等价分类——拓扑共轭18

§5 拓扑共轭与半共轭18

§5．2 Topological semi-Conjugacy and Minimal Coverings20

§5．2 拓扑半共轭与极小覆盖20

第2章 拓扑熵与混沌24

ChaPter2 Topological Entropy and Chaos24

§6 Topological Entropy25

§6 拓扑熵25

§6．1 Definition Using Open Coverings25

§6．4 Estimation and Calculation of Topological Entropy26

§6．1 拓扑熵的开覆盖定义26

§6．2 拓扑熵的Bowen定义27

§6．2 Bowen’s Definition27

§6．3 Basic Properties of Topological Entropy28

§6．2 拓扑熵的基本性质28

§6．4 拓扑熵的估计与计算36

§7．1 Two Important Theorems41

§7 混沌41

§7 Chaos41

§7．1 两个重要定理41

§7．2 Chaos in the Li-Yorke’s Sense49

§7．2 李-约克混沌49

§7．3 其他混沌52

§7．3 Other Chaos52

第8章 符号动力系统56

§8．1 Symbolic Spaces and Shifts56

§8 Symbolic Spaces and Shifts56

ChaPter3 Symbolic Dynamics56

§8 符号空间与转移自映射56

§8．1 符号空间与转移自映射56

§8．2 Dynamical Behavior of Shifts59

§8．2 符号动力系统的动力性状59

§8．3 Chaotic Behavior of Shifts62

§8．3 转移自映射的混沌性状62

§9．1 子转移和排除系统68

§9 子转移68

§9．1 Subshifts and Excluded Block Svstems68

§9 Subshifts68

§9．2 有限型子转移和阶数71

§9．2 Subshifts of Finite Type and Crder71

§10．1 斯梅尔马蹄74

§10 符号动力系统的应用74

§10．1 Smale Horseshoe74

§10 Some Applications of Symbolic Dynamics74

§10．2 转移不变集78

§10．2 Shift Invariant Sets78

§10．3 拓扑熵映射的连续性86

§10．3 The Continuity of Mappings of Topological Entropy86

§11．1 不可约性和非周期性95

Chapter4 Subshifts of Finite Type95

第4章 有限型子转移与非负方阵95

§11 非负方阵95

§11 Non-negative Matrices95

§11．1 Irreducibility and Aperiodicty95

§11．2 Directed Gragh of Non-negative Matcix100

§11．2 非负方阵的有向图100

§12 有限型子转移的转移方阵103

§12．1 Shift Matrices103

§12 Shift Matrices of Subshifts of Finite Type103

§12．1 转移方阵103

§12．2 some Restrictions on Shift Matrices105

§12．2 转移方阵的限制105

ChaPter5 Dynamical Behavior of Subshifts of Finite Type112

§13 Nonwandering sets and Transitivity112

§13．1 Nonwandering Sets112

§13．1 有限型子转移的非游荡集112

§13 有限型子转移的非游荡集与传递性112

第5章 有限型子转移的动力性状112

§13．2 Recurrence and All That119

§13．2 回复性及其他119

§14．1 子转移的拓扑熵计算126

§14 有限型子转移的拓扑熵与混沌126

§14 Topological Entropy and Chaos of Subshifts of Finite Type126

§14．1 Calculation of Topological Entropy of Subshifts126

§14．2 Chaotic Behavior of Subshifts of Finlte Type132

§14．2 有限型子转移的混沌性状132

§15 Mixing of Shifts of Finite Type137

§15 有限型子转移的混合性137

§15．1 Auxiliary Propositions138

§15．1 辅助命题138

§15．2 Some Equivalent Conditions143

§15．2 若干等价条件143

ChaPter6 General Subshifts151

第6章 一般子转移151

§16 Two examples152

§16．1 Example 1152

§16．1 例子1152

§16 两个例子152

§16．2 Example 2159

§16．2 例子2159

§17 A few substitution Dynamical Systems165

§17 代换动力系统简介165

References171

参考文献171