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

Zielgruppe

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

Autoren/Hrsg.

Vries, Jan de

Vries, Jan

Fachgebiete

• Mathematik | Informatik Mathematik Topologie
• Mathematik | Informatik Mathematik Mathematik Allgemein Diskrete Mathematik, Kombinatorik
• 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.