Vries Topological Dynamical Systems

An Introduction to the Dynamics of Continuous Mappings
1. Auflage 2014
ISBN: 978-3-11-034240-6
Verlag: De Gruyter
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 513 Seiten

Reihe: ISSN

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.

Zielgruppe

Graduate and PhD students of Mathematics, researchers; Academic libraries.

Autoren/Hrsg.

Vries, Jan de

Vries, Jan

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Diskrete Mathematik, Kombinatorik
• Mathematik | Informatik Mathematik Topologie
• Mathematik | Informatik Mathematik Geometrie Dynamische Systeme

Inhaltsverzeichnis

1;Preface;5
2;Notation;9
3;0 Introduction;17
3.1;0.1 Definition and a (very brief) historical overview;17
3.2;0.2 Continuous vs. discrete time;19
3.3;0.3 The dynamical systems point of view;23
3.4;0.4 Examples;25
4;1 Basic notions;33
4.1;1.1 Invariant and periodic points;33
4.2;1.2 Invariant sets;39
4.3;1.3 Transitivity;44
4.4;1.4 Limit sets;49
4.5;1.5 Topological conjugacy and factor mappings;51
4.6;1.6 Equicontinuity and weak mixing;60
4.7;1.7 Miscellaneous examples;73
5;2 Dynamical systems on the real line;89
5.1;2.1 Graphical iteration;89
5.2;2.2 Existence of periodic orbits;96
5.3;2.3 The truncated tent map;100
5.4;2.4 The double of a mapping;103
5.5;2.5 The Markov graph of a periodic orbit in an interval;107
5.6;2.6 Transitivity of mappings of an interval;117
6;3 Limit behaviour;133
6.1;3.1 Limit sets and attraction;133
6.2;3.2 Stability;142
6.3;3.3 Stability and attraction for periodic orbits;148
6.4;3.4 Asymptotic stability in locally compact spaces;159
6.5;3.5 The structure of (asymptotically) stable sets;169
7;4 Recurrent behaviour;181
7.1;4.1 Recurrent points;181
7.2;4.2 Almost periodic points and minimal orbit closures;185
7.3;4.3 Non-wandering points;191
7.4;4.4 Chain-recurrence;198
7.5;4.5 Asymptotic stability and basic sets;213
8;5 Shift systems;234
8.1;5.1 Notation and terminology;234
8.2;5.2 The shift mapping;239
8.3;5.3 Shift spaces;242
8.4;5.4 Factor maps;252
8.5;5.5 Subshifts and graphs;260
8.6;5.6 Recurrence, almost periodicity and mixing;269
9;6 Symbolic representations;298
9.1;6.1 Topological partitions;298
9.2;6.2 Expansive systems;309
9.3;6.3 Applications;318
10;7 Erratic behaviour;341
10.1;7.1 Stability revisited;341
10.2;7.2 Chaos(1): sensitive systems;352
10.3;7.3 Chaos(2): scrambled sets;358
10.4;7.4 Horseshoes for interval maps;371
10.5;7.5 Existence of a horseshoe;381
11;8 Topological entropy;394
11.1;8.1 The definition;394
11.2;8.2 Independence of the metric; factor maps;403
11.3;8.3 Maps on intervals and circles;407
11.4;8.4 The definition with covers;410
11.5;8.5 Miscellaneous results;418
11.6;8.6 Positive entropy and horseshoes for interval maps;422
12;A Topology;439
12.1;A.1 Elementary notions;439
12.2;A.2 Compactness;442
12.3;A.3 Continuous mappings;444
12.4;A.4 Convergence;446
12.5;A.5 Subspaces, products and quotients;448
12.6;A.6 Connectedness;450
12.7;A.7 Metric spaces;453
12.8;A.8 Baire category;460
12.9;A.9 Irreduciblemappings;462
12.10;A.10 Miscellaneous results;465
13;B The Cantor set;469
13.1;B.1 The construction;469
13.2;B.2 Proof of Brouwer’s Theorem;472
13.3;B.3 Cantor spaces;477
14;C Hints to the Exercises;481
15;Literature;497
16;Index;501

Über Autor(innen)

Jan de Vries, CWI, Amsterdam, the Netherlands.