Dynamical Systems: an Introduction
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London
Springer
2013
|
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 209 S. graph. Darst. |
| ISBN: | 9781447148357 9781447148340 |
Staff View
MARC
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| 100 | 1 | |a Barreira, Luis |d 1968- |e Verfasser |0 (DE-588)128529717 |4 aut | |
| 245 | 1 | 0 | |a Dynamical Systems |b an Introduction |c Luis Barreira ; Claudia Valls |
| 264 | 1 | |a London |b Springer |c 2013 | |
| 300 | |a X, 209 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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Record in the Search Index
| _version_ | 1804149775093727232 |
|---|---|
| adam_text | Contents
1
Introduction
................................ 1
1.1 Contents and
Suggestions
for Further Reading
........... 1
1.1.1
Basic Notions and Examples
................ 1
1.1.2
Topologica] Dynamics
.................... 2
1.1.3
Low-Dimensional Dynamics
................ 2
1.1.4
Hyperbolic Dynamics
.................... 3
1.1.5
Symbolic Dynamics
..................... 4
1.1.6
Ergodic Theory
....................... 4
1.2
Further Topics and Suggested References
.............. 5
1.3
Suggestions for Courses Based on the Book
............ 5
2
Basic Notions and Examples
....................... 7
2.1
The Notion of a Dynamical System
................. 7
2.2
Examples with Discrete Time
.................... 9
2.2.1
Rotations of the Circle
.................... 9
2.2.2
Expanding Maps of the Circle
................ 10
2.2.3
Endomorphisms of the Torus
................ 12
2.3
Examples with Continuous Time
.................. 14
2.3.1
Autonomous Differential Equations
............. 14
2.3.2
Discrete Time Versus Continuous Time
........... 18
2.3.3
Differential Equations on the
Toras
T2
........... 20
2.4
Invariant Sets
............................. 20
2.5
Exercises
............................... 24
3
Topological Dynamics
.......................... 27
3.1
Topologica!
Dynamical Systems
................... 27
3.2
Limit Sets and Basic Properties
................... 29
3.2.1
Discrete Time
........................ 29
3.2.2
Continuous Time
....................... 34
3.3
Topological Recurrence
....................... 37
3.3.1
Topological Transitivity
................... 37
viu
Contents
3.3.2
Topologica)
Mixing
..................... 39
3.4
Topological
Entropy
......................... 41
3.4.1
Basic
Notions and Examples
................ 41
3.4.2
Topologica]
Invariance
.................... 43
3.4.3
Alternative Characterizations
................ 45
3.4.4
Expansive Maps
....................... 49
3.5
Exercises
............................... 54
4
Low-Dimensional Dynamics
....................... 57
4.1
Homeomorphisms of the Circle
................... 57
4.1.1
Lifts
............................. 57
4.1.2
Rotation Number
...................... 62
4.1.3
Rational Rotation Number
.................. 64
4.1.4
Irrational Rotation Number
................. 67
4.2
Diffeomorphisms of the Circle
................... 71
4.3
Maps of the Interval
......................... 74
4.3.1
Existence of Periodic Points
................. 74
4.3.2
Sharkovsky s Theorem
................... 78
4.4
The
Poincaré-Bendixson
Theorem
................. 82
4.5
Exercises
............................... 83
5
Hyperbolic Dynamics I
87
5.1
Smooth Manifolds
.......................... 87
5.2
Hyperbolic Sets
........................... 89
5.2.)
Basic Notions
........................ 89
5.2.2
Smale
Horseshoe
...................... 91
5.2.3
Continuity of the Stable and Unstable Spaces
........ 96
5.3
Hyperbolic Sets and Invariant Families of Cones
.......... 100
5.3.1
Formulation of the Result
.................. 100
5.3.2
Existence of Invariant Families of Cones
.......... 102
5.3.3
Criterion for
Hyperbolicky
................. 105
5.4
Stability of Hyperbolic Sets
..................... 107
5.5
Exercises
............................... 109
6
Hyperbolic Dynamics II
......................... 113
6.)
Behavior Near a Hyperbolic Fixed Point
.............. 113
6.1.1
The Grobman-Hartman Theorem
.............. 113
6.1.2
The Hadamard-Perron Theorem
.............. 120
6.2
Stable and Unstable Invariant Manifolds
.............. 131
6.2.1
Existence of Invariant Manifolds
.............. 131
6.2.2
Local Product Structure
................... 135
6.3
Geodesic Flows
........................... 137
6.3.1
Hyperbolic Geometry
.................... 137
6.3.2
Quotients by Isometries
................... 142
6.3.3
Geodesic Flow
........................ 144
6.3.4
Hyperbolic Flows
...................... 147
6.4
Exercises
............................... 150
Contents
їх
7
Symbolic
Dynamics............................ 153
7.1
Basic Notions
............................ 153
7.1.1
Shift Map
.......................... 153
7.1.2
Topologica)
Entropy
..................... 155
7.1.3
Two-Sided Sequences
.................... 156
7.2
Examples of Codings
........................ 157
7.2.1
Expanding Maps
....................... 157
7.2.2
Quadratic Maps
....................... 161
7.2.3
The
Smale
Horseshoe
.................... 162
7.3
Topological Markov Chains
..................... 164
7.3.1
Basic Notions
........................ 164
7.3.2
Periodic Points
........................ 165
7.3.3
Topological Entropy
..................... 167
7.3.4
Topologica]
Recurrence
................... 169
7.4
Horseshoes and
Topological
Markov Chains
............ 172
7.5
7,eta Functions
............................ 175
7.6
Exercises
............................... 178
8
Ergodic Theory
.............................. 181
8.1
Notions from Measure Theory
.................... 181
8.2
Invariant Measures
.......................... 183
8.3 Nontrivial
Recurrence
........................ 188
8.4
The Ergodic Theorem
........................ 189
8.5
Metric Entropy
............................ 191
8.6
Proof of the Ergodic Theorem
.................... 194
8.7
Exercises
............................... 200
References
................................... 203
Index
...................................... 207
|
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| isbn | 9781447148357 9781447148340 |
| language | English |
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| physical | X, 209 S. graph. Darst. |
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| spelling | Barreira, Luis 1968- Verfasser (DE-588)128529717 aut Dynamical Systems an Introduction Luis Barreira ; Claudia Valls London Springer 2013 X, 209 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Dynamisches System (DE-588)4013396-5 gnd rswk-swf Ergodentheorie (DE-588)4015246-7 gnd rswk-swf Dynamisches System (DE-588)4013396-5 s Ergodentheorie (DE-588)4015246-7 s DE-604 Valls, Claudia 1973- Sonstige (DE-588)1026700426 oth Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025481057&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Barreira, Luis 1968- Dynamical Systems an Introduction Dynamisches System (DE-588)4013396-5 gnd Ergodentheorie (DE-588)4015246-7 gnd |
| subject_GND | (DE-588)4013396-5 (DE-588)4015246-7 |
| title | Dynamical Systems an Introduction |
| title_auth | Dynamical Systems an Introduction |
| title_exact_search | Dynamical Systems an Introduction |
| title_full | Dynamical Systems an Introduction Luis Barreira ; Claudia Valls |
| title_fullStr | Dynamical Systems an Introduction Luis Barreira ; Claudia Valls |
| title_full_unstemmed | Dynamical Systems an Introduction Luis Barreira ; Claudia Valls |
| title_short | Dynamical Systems |
| title_sort | dynamical systems an introduction |
| title_sub | an Introduction |
| topic | Dynamisches System (DE-588)4013396-5 gnd Ergodentheorie (DE-588)4015246-7 gnd |
| topic_facet | Dynamisches System Ergodentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025481057&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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