Topological Dynamical Systems (eBook)

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Jan de Vries, CWI, Amsterdam, the Netherlands.

Preface 5
Notation 9
0 Introduction 17
0.1 Definition and a (very brief) historical overview 17
0.2 Continuous vs. discrete time 19
0.3 The dynamical systems point of view 23
0.4 Examples 25
1 Basic notions 33
1.1 Invariant and periodic points 33
1.2 Invariant sets 39
1.3 Transitivity 44
1.4 Limit sets 49
1.5 Topological conjugacy and factor mappings 51
1.6 Equicontinuity and weak mixing 60
1.7 Miscellaneous examples 73
2 Dynamical systems on the real line 89
2.1 Graphical iteration 89
2.2 Existence of periodic orbits 96
2.3 The truncated tent map 100
2.4 The double of a mapping 103
2.5 The Markov graph of a periodic orbit in an interval 107
2.6 Transitivity of mappings of an interval 117
3 Limit behaviour 133
3.1 Limit sets and attraction 133
3.2 Stability 142
3.3 Stability and attraction for periodic orbits 148
3.4 Asymptotic stability in locally compact spaces 159
3.5 The structure of (asymptotically) stable sets 169
4 Recurrent behaviour 181
4.1 Recurrent points 181
4.2 Almost periodic points and minimal orbit closures 185
4.3 Non-wandering points 191
4.4 Chain-recurrence 198
4.5 Asymptotic stability and basic sets 213
5 Shift systems 234
5.1 Notation and terminology 234
5.2 The shift mapping 239
5.3 Shift spaces 242
5.4 Factor maps 252
5.5 Subshifts and graphs 260
5.6 Recurrence, almost periodicity and mixing 269
6 Symbolic representations 298
6.1 Topological partitions 298
6.2 Expansive systems 309
6.3 Applications 318
7 Erratic behaviour 341
7.1 Stability revisited 341
7.2 Chaos(1): sensitive systems 352
7.3 Chaos(2): scrambled sets 358
7.4 Horseshoes for interval maps 371
7.5 Existence of a horseshoe 381
8 Topological entropy 394
8.1 The definition 394
8.2 Independence of the metric factor maps
8.3 Maps on intervals and circles 407
8.4 The definition with covers 410
8.5 Miscellaneous results 418
8.6 Positive entropy and horseshoes for interval maps 422
A Topology 439
A.1 Elementary notions 439
A.2 Compactness 442
A.3 Continuous mappings 444
A.4 Convergence 446
A.5 Subspaces, products and quotients 448
A.6 Connectedness 450
A.7 Metric spaces 453
A.8 Baire category 460
A.9 Irreduciblemappings 462
A.10 Miscellaneous results 465
B The Cantor set 469
B.1 The construction 469
B.2 Proof of Brouwer’s Theorem 472
B.3 Cantor spaces 477
C Hints to the Exercises 481
Literature 497
Index 501

lt;P>"It [the book] is well organized, with careful proofs and many examples, figures, and exercises (with hints), along with end-of-chapter notes providing historical context and motivation, and an extensive index. [...] This book could serve as a textbook for an introductory course in topological dynamics or as a supplement to a more general dynamical systems course. For researchers, it is very useful to have so many results gathered and organized in one place, especially the more general results on the non-compact, non-metric, and non-invertible cases. This book is a valuable contribution, both as a text and as a reference." Mathematical Reviews

"Students who have mastered this book will have a solid basis to start research on related topics. [...] This book is worth reading to start research in topological dynamics." Zentralblatt für Mathematik