A first course in ergodic theory:
"A First Course in Ergodic Theory provides readers with an introductory course in Ergodic Theory. This textbook has been developed from the authors' own notes on the subject, which they have been teaching since the 1990s. Over the years they have added topics, theorems, examples and explan...
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boca Raton ; London ; New York
CRC Press
2021
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| Ausgabe: | First edition |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | "A First Course in Ergodic Theory provides readers with an introductory course in Ergodic Theory. This textbook has been developed from the authors' own notes on the subject, which they have been teaching since the 1990s. Over the years they have added topics, theorems, examples and explanations from various sources. The result is a book that is easy to teach from and easy to learn from - designed to require only minimal prerequisites. Features Suitable for readers with only a basic knowledge of measure theory, some topology and a very basic knowledge of functional analysis Perfect as the primary textbook for a course in Ergodic Theory Examples are described and are studied in detail when new properties are presented"-- |
| Beschreibung: | xiii, 253 Seiten Illustrationen |
| ISBN: | 9780367226206 9781032021843 |
Internformat
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| 520 | 3 | |a "A First Course in Ergodic Theory provides readers with an introductory course in Ergodic Theory. This textbook has been developed from the authors' own notes on the subject, which they have been teaching since the 1990s. Over the years they have added topics, theorems, examples and explanations from various sources. The result is a book that is easy to teach from and easy to learn from - designed to require only minimal prerequisites. Features Suitable for readers with only a basic knowledge of measure theory, some topology and a very basic knowledge of functional analysis Perfect as the primary textbook for a course in Ergodic Theory Examples are described and are studied in detail when new properties are presented"-- | |
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Datensatz im Suchindex
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| adam_text | Contents Preface xi Author Bios Chapter 1.1 1.2 1.3 Chapter 1 ■ Measure Preservingness and Basic Examples xiii 1 WHAT IS ERGODIC THEORY? MEASURE PRESERVING TRANSFORMATIONS BASIC EXAMPLES 1 3 6 2 ■ Recurrence and Ergodicity_____________________ 19 2.1 RECURRENCE 19 2.2 ERGODICITY 22 2.3 EXAMPLES OF ERGODIC TRANSFORMATIONS 27 Chapter 3 ■ The Pointwise Ergodic Theorem and Mixing 33 3.1 THE POINTWISE ERGODIC THEOREM 34 3.2 NORMAL NUMBERS 43 3.3 IRREDUCIBLE MARKOV CHAINS 46 3.4 MIXING 49 Chapter 4 ■ More Ergodic Theorems 53 4.1 THE MEAN ERGODIC THEOREM 53 4.2 THE HUREWICZ ERGODIC THEOREM 56 vii
viii ■ Contents Chapter 5 ■ Isomorphisms and Factor Maps 65 5.1 MEASURE PRESERVING ISOMORPHISMS 65 5.2 FACTOR MAPS 71 5.3 NATURAL EXTENSIONS 72 Chapter 6 ■ The Perron-Frobenius Operator 77 6.1 ABSOLUTELY CONTINUOUS INVARIANT MEASURES 77 6.2 EXACTNESS 82 6.3 PIECEWISE MONOTONE INTERVAL MAPS 91 Chapter 7 ■ Invariant Measures for Continuous Transforma tions 101 7.1 EXISTENCE 101 7.2 UNIQUE ERGODICITY AND UNIFORM DISTRIBUTION 109 7.3 SOME TOPOLOGICAL DYNAMICS 115 Chapter 8 ■ Continued Fractions 125 8.1 REGULAR CONTINUED FRACTIONS 126 8.2 ERGODIC PROPERTIES OF THE GAUSS MAP 130 8.3 THE DOEBLIN-LENSTRA CONJECTURE 135 8.4 OTHER CONTINUED FRACTION TRANSFORMATIONS 139 Chapter 9 ■ Entropy____________________________ 147 9.1 RANDOMNESS AND INFORMATION 147 9.2 DEFINITIONS AND PROPERTIES 150 9.3 CALCULATION OF ENTROPY AND EXAMPLES 156 9.4 THE SHANNON-MCMILLAN-BREIMAN THEOREM 160 9.5 LOCHS’THEOREM 168
Contents ■ ix Chapter 10.1 10.2 10.3 Chapter 11.1 11.2 10 ■ The Variational Principle 173 TOPOLOGICAL ENTROPY 173 PROOF OF THE VARIATIONALPRINCIPLE MEASURES OF MAXIMAL ENTROPY 11· Infinite Ergodic Theory EXAMPLES CONSERVATIVE AND DISSIPATIVE PART 188 194 199 199 202 11.3 INDUCED SYSTEMS 207 11.4 JUMP TRANSFORMATIONS 216 11.5 INFINITE ERGODIC THEOREMS 218 Chapter 12 »Appendix 223 12.1 TOPOLOGY 223 12.2 MEASURE THEORY 224 12.3 LEBESGUE SPACES 229 12.4 LEBESGUE INTEGRATION 231 12.5 HILBERT SPACES 233 12.6 BOREL MEASURES ON COMPACT METRIC SPACES 235 12.7 FUNCTIONS OF BOUNDED VARIATION 237 Bibliography 241 Index 247
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| adam_txt |
Contents Preface xi Author Bios Chapter 1.1 1.2 1.3 Chapter 1 ■ Measure Preservingness and Basic Examples xiii 1 WHAT IS ERGODIC THEORY? MEASURE PRESERVING TRANSFORMATIONS BASIC EXAMPLES 1 3 6 2 ■ Recurrence and Ergodicity_ 19 2.1 RECURRENCE 19 2.2 ERGODICITY 22 2.3 EXAMPLES OF ERGODIC TRANSFORMATIONS 27 Chapter 3 ■ The Pointwise Ergodic Theorem and Mixing 33 3.1 THE POINTWISE ERGODIC THEOREM 34 3.2 NORMAL NUMBERS 43 3.3 IRREDUCIBLE MARKOV CHAINS 46 3.4 MIXING 49 Chapter 4 ■ More Ergodic Theorems 53 4.1 THE MEAN ERGODIC THEOREM 53 4.2 THE HUREWICZ ERGODIC THEOREM 56 vii
viii ■ Contents Chapter 5 ■ Isomorphisms and Factor Maps 65 5.1 MEASURE PRESERVING ISOMORPHISMS 65 5.2 FACTOR MAPS 71 5.3 NATURAL EXTENSIONS 72 Chapter 6 ■ The Perron-Frobenius Operator 77 6.1 ABSOLUTELY CONTINUOUS INVARIANT MEASURES 77 6.2 EXACTNESS 82 6.3 PIECEWISE MONOTONE INTERVAL MAPS 91 Chapter 7 ■ Invariant Measures for Continuous Transforma tions 101 7.1 EXISTENCE 101 7.2 UNIQUE ERGODICITY AND UNIFORM DISTRIBUTION 109 7.3 SOME TOPOLOGICAL DYNAMICS 115 Chapter 8 ■ Continued Fractions 125 8.1 REGULAR CONTINUED FRACTIONS 126 8.2 ERGODIC PROPERTIES OF THE GAUSS MAP 130 8.3 THE DOEBLIN-LENSTRA CONJECTURE 135 8.4 OTHER CONTINUED FRACTION TRANSFORMATIONS 139 Chapter 9 ■ Entropy_ 147 9.1 RANDOMNESS AND INFORMATION 147 9.2 DEFINITIONS AND PROPERTIES 150 9.3 CALCULATION OF ENTROPY AND EXAMPLES 156 9.4 THE SHANNON-MCMILLAN-BREIMAN THEOREM 160 9.5 LOCHS’THEOREM 168
Contents ■ ix Chapter 10.1 10.2 10.3 Chapter 11.1 11.2 10 ■ The Variational Principle 173 TOPOLOGICAL ENTROPY 173 PROOF OF THE VARIATIONALPRINCIPLE MEASURES OF MAXIMAL ENTROPY 11· Infinite Ergodic Theory EXAMPLES CONSERVATIVE AND DISSIPATIVE PART 188 194 199 199 202 11.3 INDUCED SYSTEMS 207 11.4 JUMP TRANSFORMATIONS 216 11.5 INFINITE ERGODIC THEOREMS 218 Chapter 12 »Appendix 223 12.1 TOPOLOGY 223 12.2 MEASURE THEORY 224 12.3 LEBESGUE SPACES 229 12.4 LEBESGUE INTEGRATION 231 12.5 HILBERT SPACES 233 12.6 BOREL MEASURES ON COMPACT METRIC SPACES 235 12.7 FUNCTIONS OF BOUNDED VARIATION 237 Bibliography 241 Index 247 |
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| author | Dajani, Karma 1958- Kalle, Charlene 1979- |
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| discipline | Mathematik |
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| id | DE-604.BV047226287 |
| illustrated | Illustrated |
| index_date | 2024-07-03T16:59:17Z |
| indexdate | 2024-07-10T09:06:13Z |
| institution | BVB |
| isbn | 9780367226206 9781032021843 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-032631018 |
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| physical | xiii, 253 Seiten Illustrationen |
| publishDate | 2021 |
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| publisher | CRC Press |
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| spelling | Dajani, Karma 1958- Verfasser (DE-588)1238264913 aut A first course in ergodic theory Karma Dajani (Utrecht University), Charlene Kalle (Leiden University) First edition Boca Raton ; London ; New York CRC Press 2021 xiii, 253 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier "A First Course in Ergodic Theory provides readers with an introductory course in Ergodic Theory. This textbook has been developed from the authors' own notes on the subject, which they have been teaching since the 1990s. Over the years they have added topics, theorems, examples and explanations from various sources. The result is a book that is easy to teach from and easy to learn from - designed to require only minimal prerequisites. Features Suitable for readers with only a basic knowledge of measure theory, some topology and a very basic knowledge of functional analysis Perfect as the primary textbook for a course in Ergodic Theory Examples are described and are studied in detail when new properties are presented"-- Ergodentheorie (DE-588)4015246-7 gnd rswk-swf Ergodic theory Ergodentheorie (DE-588)4015246-7 s DE-604 Kalle, Charlene 1979- Verfasser (DE-588)1049948068 aut Erscheint auch als Online-Ausgabe 978-0-429-27601-9 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032631018&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dajani, Karma 1958- Kalle, Charlene 1979- A first course in ergodic theory Ergodentheorie (DE-588)4015246-7 gnd |
| subject_GND | (DE-588)4015246-7 |
| title | A first course in ergodic theory |
| title_auth | A first course in ergodic theory |
| title_exact_search | A first course in ergodic theory |
| title_exact_search_txtP | A first course in ergodic theory |
| title_full | A first course in ergodic theory Karma Dajani (Utrecht University), Charlene Kalle (Leiden University) |
| title_fullStr | A first course in ergodic theory Karma Dajani (Utrecht University), Charlene Kalle (Leiden University) |
| title_full_unstemmed | A first course in ergodic theory Karma Dajani (Utrecht University), Charlene Kalle (Leiden University) |
| title_short | A first course in ergodic theory |
| title_sort | a first course in ergodic theory |
| topic | Ergodentheorie (DE-588)4015246-7 gnd |
| topic_facet | Ergodentheorie |
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