Introduction to dynamical systems

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Introduction......Page 13
1.1 The Notion of a Dynamical System......Page 15
1.2 Circle Rotations......Page 17
1.3 Expanding Endomorphisms of the Circle......Page 19
1.4 Shifts and Subshifts......Page 21
1.5 Quadratic Maps......Page 23
1.6 The Gauss Transformation......Page 25
1.7 Hyperbolic Toral Automorphisms......Page 27
1.8 The Horseshoe......Page 29
1.9 The Solenoid......Page 31
1.10 Flows and Differential Equations......Page 33
1.11 Suspension and Cross-Section......Page 35
1.12 Chaos and Lyapunov Exponents......Page 37
1.13 Attractors......Page 39
2.1 Limit Sets and Recurrence......Page 42
2.2 Topological Transitivity......Page 45
2.3 Topological Mixing......Page 47
2.4 Expansiveness......Page 49
2.5 Topological Entropy......Page 50
2.6 Topological Entropy for Some Examples......Page 55
2.7 Equicontinuity, Distality, and Proximality......Page 59
2.8 Applications of Topological Recurrence to Ramsey Theory......Page 62
CHAPTER THREE Symbolic Dynamics......Page 68
3.1 Subshifts and Codes......Page 69
3.2 Subshifts of Finite Type......Page 70
3.3 The Perron–Frobenius Theorem......Page 71
3.4 Topological Entropy and the Zeta Function of an SFT......Page 74
3.5 Strong Shift Equivalence and Shift Equivalence......Page 76
3.6 Substitutions......Page 78
3.7 Sofic Shifts......Page 80
3.8 Data Storage......Page 81
4.1 Measure-Theory Preliminaries......Page 83
4.2 Recurrence......Page 85
4.3 Ergodicity and Mixing......Page 87
4.4 Examples......Page 91
4.5 Ergodic Theorems......Page 94
4.6 Invariant Measures for Continuous Maps......Page 99
4.7 Unique Ergodicity and Weyl’s Theorem......Page 101
4.8 The Gauss Transformation Revisited......Page 104
4.9 Discrete Spectrum......Page 108
4.10 Weak Mixing......Page 111
4.11 Applications of Measure-Theoretic Recurrence to Number Theory......Page 115
4.12 Internet Search......Page 117
CHAPTER FIVE Hyperbolic Dynamics......Page 120
5.1 Expanding Endomorphisms Revisited......Page 121
5.2 Hyperbolic Sets......Page 122
5.3 -Orbits......Page 124
5.4 Invariant Cones......Page 128
5.5 Stability of Hyperbolic Sets......Page 131
5.6 Stable and Unstable Manifolds......Page 132
5.7 Inclination Lemma......Page 136
5.8 Horseshoes and Transverse Homoclinic Points......Page 138
5.9 Local Product Structure and Locally Maximal Hyperbolic Sets......Page 142
5.10 Anosov Diffeomorphisms......Page 144
5.11 Axiom A and Structural Stability......Page 147
5.12 Markov Partitions......Page 148
5.13 Appendix: Differentiable Manifolds......Page 151
6.1 Hölder Continuity of the Stable and Unstable Distributions......Page 155
6.2 Absolute Continuity of the Stable and Unstable Foliations......Page 158
6.3 Proof of Ergodicity......Page 165
7.1 Circle Homeomorphisms......Page 167
7.2 Circle Diffeomorphisms......Page 174
7.3 The Sharkovsky Theorem......Page 176
7.4 Combinatorial Theory of Piecewise-Monotone Mappings......Page 184
7.5 The Schwarzian Derivative......Page 192
7.6 Real Quadratic Maps......Page 195
7.7 Bifurcations of Periodic Points......Page 197
7.8 The Feigenbaum Phenomenon......Page 203
8.1 Complex Analysis on the Riemann Sphere......Page 205
8.2 Examples......Page 208
8.3 Normal Families......Page 211
8.4 Periodic Points......Page 212
8.5 The Julia Set......Page 214
8.6 The Mandelbrot Set......Page 219
9.1 Entropy of a Partition......Page 222
9.2 Conditional Entropy......Page 225
9.3 Entropy of a Measure-Preserving Transformation......Page 227
9.4 Examples of Entropy Calculation......Page 232
9.5 Variational Principle......Page 235
Bibliography......Page 239
Index......Page 245