Randomness and recurrence in dynamical systems: a real analysis approach
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Washington, DC
MAA
2010
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| Series: | The Carus mathematical monographs
31 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 357 S. graph. Darst. |
| ISBN: | 9780883850435 |
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| adam_text | Titel: Randomness and recurrence in dynamical systems
Autor: Nillsen, Rodney Victor
Jahr: 2010
Contents
Foreword xiii
Preface xv
1 Background Ideas and Knowledge 1
1.1 Dynamical systems, iteration and orbits......... 1
1.2 Information loss and randomness in dynamical systems . 4
1.3 Assumed knowledge and notation............ 10
Appendix: Mathematical reasoning and proof......... 12
Exercises ............................ 17
Investigations.......................... 19
Notes .............................. 19
Bibliography .......................... 22
2 Irrational Numbers and Dynamical Systems 25
2.1 Introduction: irrational numbers and the infinite..... 25
2.2 Fractional parts and points on the unit circle....... 28
2.3 Partitions and the Pigeon-hole Principle......... 33
2.4 Kronecker s Theorem................... 34
2.5 The dynamical systems approach to Kronecker s Theorem 41
2.6 Kronecker and chaos in the music of Steve Reich .... 46
2.7 The ideas in Weyl s Theorem on irrational numbers ... 52
2.8 The proof of Weyl s Theorem............... 57
2.9 Chaos in Kronecker systems............... 70
Exercises ............................ 75
IX
CONTENTS
Investigations.......................... 80
Notes .............................. 83
Bibliography .......................... 85
Probability and Randomness 87
3.1 Introduction: probability, coin tossing and randomness . 87
3.2 Expansions to a base ................... 92
3.3 Rational numbers and periodic expansions........ 101
3.4 Sets, events, length and probability............ 106
3.5 Sets of measure zero ................... 117
3.6 Independent sets and events................ 122
3.7 Typewriters, recurrence, and the Prince of Denmark ... 125
3.8 The Rademacher functions................ 136
3.9 Randomness, binary expansions and a law of averages . 142
3.10 The dynamical systems approach............. 148
3.11 The Walsh functions ................... 158
3.12 Normal numbers and randomness ............ 163
3.13 Notions of probability and randomness.......... 173
3.14 The curious phenomenon of the leading significant digit. 180
3.15 Leading digits and geometric sequences......... 186
3.16 Multiple digits and a result of Diaconis.......... 193
3.17 Dynamical systems and changes of scale......... 198
3.18 The equivalence of Kronecker and Benford systems . . . 205
3.19 Scale invariance and the necessity of Benford s law . . . 208
Exercises ............................ 220
Investigations.......................... 225
Notes .............................. 227
Bibliography .......................... 230
Recurrence 235
4.1 Introduction: random systems and recurrence...... 235
4.2 Transformations that preserve length........... 239
4.3 Poincare recurrence.................... 248
4.4 Recurrent points...................... 253
4.5 Kac s result on average recurrence times......... 256
CONTENTS xi
4.6 Applications to the Kronecker and Borel systems .... 266
4.7 The standard deviation of recurrence times........ 270
Exercises ............................ 283
Investigations.......................... 284
Notes .............................. 286
Bibliography .......................... 287
5 Averaging in Time and Space 289
5.1 Introduction: averaging in time and space........ 289
5.2 Outer measure....................... 293
5.3 Invariant sets ....................... 300
5.4 Measurable sets...................... 303
5.5 Measure-preserving transformations........... 307
5.6 Poincare recurrence ... again............... 310
5.7 Ergodic systems...................... 312
5.8 Birkhoff s Theorem on time and space averages..... 318
5.9 Weyl s Theorem from the ergodic viewpoint....... 320
5.10 The Ergodic Theorem and expansions to an arbitrary base 323
5.11 Kac s recurrence formula: the general case........ 327
5.12 Mixing transformations and an example of Kakutani . . 328
5.13 Liiroth transformations and continued fractions..... 333
Exercises ............................ 342
Investigations.......................... 345
Notes .............................. 346
Bibliography .......................... 347
Index of Subjects 351
Index of Symbols 355
About the Author 357
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| indexdate | 2024-07-09T22:54:47Z |
| institution | BVB |
| isbn | 9780883850435 |
| language | English |
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| physical | XVIII, 357 S. graph. Darst. |
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| spelling | Nillsen, Rodney Victor 1945- Verfasser (DE-588)114432473 aut Randomness and recurrence in dynamical systems a real analysis approach Rodney Nillsen Washington, DC MAA 2010 XVIII, 357 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier The Carus mathematical monographs 31 Zufall (DE-588)4068050-2 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Dynamisches System (DE-588)4013396-5 s Zufall (DE-588)4068050-2 s DE-604 The Carus mathematical monographs 31 (DE-604)BV001887236 31 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021178469&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nillsen, Rodney Victor 1945- Randomness and recurrence in dynamical systems a real analysis approach The Carus mathematical monographs Zufall (DE-588)4068050-2 gnd Dynamisches System (DE-588)4013396-5 gnd |
| subject_GND | (DE-588)4068050-2 (DE-588)4013396-5 |
| title | Randomness and recurrence in dynamical systems a real analysis approach |
| title_auth | Randomness and recurrence in dynamical systems a real analysis approach |
| title_exact_search | Randomness and recurrence in dynamical systems a real analysis approach |
| title_full | Randomness and recurrence in dynamical systems a real analysis approach Rodney Nillsen |
| title_fullStr | Randomness and recurrence in dynamical systems a real analysis approach Rodney Nillsen |
| title_full_unstemmed | Randomness and recurrence in dynamical systems a real analysis approach Rodney Nillsen |
| title_short | Randomness and recurrence in dynamical systems |
| title_sort | randomness and recurrence in dynamical systems a real analysis approach |
| title_sub | a real analysis approach |
| topic | Zufall (DE-588)4068050-2 gnd Dynamisches System (DE-588)4013396-5 gnd |
| topic_facet | Zufall Dynamisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021178469&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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