Topological Dynamical Systems
Description
There is no recent elementary introduction to the theory of discrete dynamical systems that stresses the topological background of the topic. This book fills this gap: it deals with this theory as 'applied general topology'. We treat all important concepts needed to understand recent literature. The book is addressed primarily to graduate students. The prerequisites for understanding this book are modest: a certain mathematical maturity and course in General Topology are sufficient.
Reviews / Votes
"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
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Content
2 - Notation [Seite 9]
3 - 0 Introduction [Seite 17]
3.1 - 0.1 Definition and a (very brief) historical overview [Seite 17]
3.2 - 0.2 Continuous vs. discrete time [Seite 19]
3.3 - 0.3 The dynamical systems point of view [Seite 23]
3.4 - 0.4 Examples [Seite 25]
4 - 1 Basic notions [Seite 33]
4.1 - 1.1 Invariant and periodic points [Seite 33]
4.2 - 1.2 Invariant sets [Seite 39]
4.3 - 1.3 Transitivity [Seite 44]
4.4 - 1.4 Limit sets [Seite 49]
4.5 - 1.5 Topological conjugacy and factor mappings [Seite 51]
4.6 - 1.6 Equicontinuity and weak mixing [Seite 60]
4.7 - 1.7 Miscellaneous examples [Seite 73]
5 - 2 Dynamical systems on the real line [Seite 89]
5.1 - 2.1 Graphical iteration [Seite 89]
5.2 - 2.2 Existence of periodic orbits [Seite 96]
5.3 - 2.3 The truncated tent map [Seite 100]
5.4 - 2.4 The double of a mapping [Seite 103]
5.5 - 2.5 The Markov graph of a periodic orbit in an interval [Seite 107]
5.6 - 2.6 Transitivity of mappings of an interval [Seite 117]
6 - 3 Limit behaviour [Seite 133]
6.1 - 3.1 Limit sets and attraction [Seite 133]
6.2 - 3.2 Stability [Seite 142]
6.3 - 3.3 Stability and attraction for periodic orbits [Seite 148]
6.4 - 3.4 Asymptotic stability in locally compact spaces [Seite 159]
6.5 - 3.5 The structure of (asymptotically) stable sets [Seite 169]
7 - 4 Recurrent behaviour [Seite 181]
7.1 - 4.1 Recurrent points [Seite 181]
7.2 - 4.2 Almost periodic points and minimal orbit closures [Seite 185]
7.3 - 4.3 Non-wandering points [Seite 191]
7.4 - 4.4 Chain-recurrence [Seite 198]
7.5 - 4.5 Asymptotic stability and basic sets [Seite 213]
8 - 5 Shift systems [Seite 234]
8.1 - 5.1 Notation and terminology [Seite 234]
8.2 - 5.2 The shift mapping [Seite 239]
8.3 - 5.3 Shift spaces [Seite 242]
8.4 - 5.4 Factor maps [Seite 252]
8.5 - 5.5 Subshifts and graphs [Seite 260]
8.6 - 5.6 Recurrence, almost periodicity and mixing [Seite 269]
9 - 6 Symbolic representations [Seite 298]
9.1 - 6.1 Topological partitions [Seite 298]
9.2 - 6.2 Expansive systems [Seite 309]
9.3 - 6.3 Applications [Seite 318]
10 - 7 Erratic behaviour [Seite 341]
10.1 - 7.1 Stability revisited [Seite 341]
10.2 - 7.2 Chaos(1): sensitive systems [Seite 352]
10.3 - 7.3 Chaos(2): scrambled sets [Seite 358]
10.4 - 7.4 Horseshoes for interval maps [Seite 371]
10.5 - 7.5 Existence of a horseshoe [Seite 381]
11 - 8 Topological entropy [Seite 394]
11.1 - 8.1 The definition [Seite 394]
11.2 - 8.2 Independence of the metric [Seite 403]
11.3 - 8.3 Maps on intervals and circles [Seite 407]
11.4 - 8.4 The definition with covers [Seite 410]
11.5 - 8.5 Miscellaneous results [Seite 418]
11.6 - 8.6 Positive entropy and horseshoes for interval maps [Seite 422]
12 - A Topology [Seite 439]
12.1 - A.1 Elementary notions [Seite 439]
12.2 - A.2 Compactness [Seite 442]
12.3 - A.3 Continuous mappings [Seite 444]
12.4 - A.4 Convergence [Seite 446]
12.5 - A.5 Subspaces, products and quotients [Seite 448]
12.6 - A.6 Connectedness [Seite 450]
12.7 - A.7 Metric spaces [Seite 453]
12.8 - A.8 Baire category [Seite 460]
12.9 - A.9 Irreduciblemappings [Seite 462]
12.10 - A.10 Miscellaneous results [Seite 465]
13 - B The Cantor set [Seite 469]
13.1 - B.1 The construction [Seite 469]
13.2 - B.2 Proof of Brouwer's Theorem [Seite 472]
13.3 - B.3 Cantor spaces [Seite 477]
14 - C Hints to the Exercises [Seite 481]
15 - Literature [Seite 497]
16 - Index [Seite 501]
