Operator theoretic aspects of ergodic theory:
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| Main Authors: | , , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham [u.a.]
Springer
2015
|
| Series: | Graduate texts in mathematics
272 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 577 - 611 |
| Physical Description: | XVIII, 628 S. Ill. |
| ISBN: | 9783319168975 |
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MARC
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| 245 | 1 | 0 | |a Operator theoretic aspects of ergodic theory |c Tanja Eisner ; Bálint Farkas ; Markus Haase ; Rainer Nagel |
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Record in the Search Index
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| adam_text | Contents Preface.................................................................................................................... vii 1 What Is Ergodic Theory?............................................................................ Exercises ........................................................................................................ 1 6 2 Topological Dynamical Systems................................................................. 2.1 Basic Examples................................................................................... 2.2 Basic Constructions............................................................................. 2.3 Topological Transitivity..................................................................... 2.4 Transitivity of Subshifts..................................................................... Exercises........................................................................................................ 9 10 15 22 26 29 3 Minimality and Recurrence....................................................................... 3.1 Minimality ........................................................................................... 3.2 Topological Recurrence ..................................................................... 3.3 Recurrence in Extensions.................................................................... Exercises........................................................................................................ 33 34 36 39 42 4 The C*-Algebra C(Æ) and the KoopmanOperator................................. 4.1
Continuous Functions on CompactSpaces....................................... 4.2 The Space C(K) as a Commutative C*-Algebra.............................. 4.3 The Koopman Operator...................................................................... 4.4 The Gelfand-Naimark Theorem....................................................... Supplement: Proof of the Gelfand-Naimark Theorem.............................. Exercises........................................................................................................ 45 46 51 54 59 60 68 5 Measure-Preserving Systems...................................................................... 5.1 Examples............................................................................................. 5.2 Measures on Compact Spaces............................................................ 5.3 Haar Measures and Rotations............................................................ Supplement: Haar Measures on Homogeneous Spaces............................. Exercises........................................................................................................ 71 73 79 84 87 90 XV
xvi 6 Contents Recurrence and Ergodicity........................................................................... 6.1 The Measure Algebra and Invertible Systems................................... 6.2 Recurrence............................................................................................ 6.3 Ergodicity............................................................................................. Supplement: Induced Transformation and the Kakutani-Rokhlin Lemma..................................................... 107 Exercises........................................................................................................ 7 The Banach Lattice Մ and the Koopman Operator................................ 7.1 Banach Lattices and Positive Operators............................................ 7.2 The Space L(X) as a Banach Lattice................................................. 7.3 The Koopman Operator and Ergodicity.............................................. Supplement: Interplay Between Lattice and Algebra Structure............... Exercises ........................................................................................................ 8 The Mean Ergodic Theorem.......................................................................... 8.1 Von Neumann’s Mean Ergodic Theorem........................................... 8.2 The Fixed Space and Ergodicity......................................................... 8.3 Perron’s Theorem and the Ergodicity of Markov Shifts.................. 8.4 Mean Ergodic
Operators..................................................................... 8.5 Operations with Mean Ergodic Operators......................................... Supplement: Mean Ergodic Operator Semigroups..................................... Exercises........................................................................................................ 9 Mixing Dynamical Systems............................................................................. 9.1 Strong Mixing....................................................................................... 9.2 Weak Mixing........................................................................................ 9.3 More Characterizations of Weakly Mixing Systems......................... 9.4 Weak Mixing of All Orders................................................................ Exercises........................................................................................................ 95 95 99 104 110 115 117 122 124 129 132 135 136 140 143 145 150 152 158 161 162 172 179 182 187 191 10.1 Invariant Measures.............................................................................. 192 10.2 Uniquely and Strictly Ergodic Systems.............................................. 194 10.3 Mean Ergodicity of Group Rotations................................................ 197 10.4 Furstenberg’s Theorem on Group Extensions.................................. 199 10.5 Application: Equidistribution ............................................................ 203 Exercises
........................................................................................................ 206 10 Mean Ergodic Operators on C(K)................................................................ 11 The Pointwise Ergodic Theorem................................................................... 11.1 Pointwise Ergodic Operators............................................................. 11.2 Banach’s Principle and Maximal Inequalities................................... 11.3 Applications......................................................................................... Exercises........................................................................................................ 211 212 214 218 222 225 12.1 Point Isomorphisms and Factor Maps.............................................. 226 12.2 Algebra Isomorphisms of Measure-Preserving Systems................. 230 12.3 Abstract Systems and Topological Models....................................... 235 12 Isomorphisms and Topological Models ......................................................
Contents xvii 12.4 The Stone Representation.................................................................. 240 12.5 Mean Ergodicity on Subalgebras....................................................... 242 Exercises........................................................................................................ 245 13 Markov Operators...................................................................................... 13.1 Examples and Basic Properties.......................................................... 13.2 Embeddings, Factor Maps and Isomorphisms.................................. 13.3 Markov Projections............................................................................ 13.4 Factors and Topological Models....................................................... 13.5 Inductive Limits and the Invertible Extension.................................. Exercises........................................................................................................ 249 250 254 257 262 265 271 14 Compact Groups ........................................................................................ 14.1 Compact Groups and the Haar Measure........................................... 14.2 The Character Group......................................................................... 14.3 The Pontryagin Duality Theorem..................................................... 14.4 Monothetic Groups and Ergodic Rotations...................................... Supplement: Characters of Discrete Abelian Groups................................
Exercises........................................................................................................ 273 273 276 279 283 287 288 15 Group Actions and Representations ...................................................... 15.1 Continuous Representations on BanachSpaces.............................. 15.2 Unitary Representations................................................................... 15.3 Compact Group Actions................................................................... 15.4 Markov Representations................................................................... Supplement: Abstract Compact Group Extensions.................................... Exercises........................................................................................................ 291 292 297 303 306 308 312 16 The Jacobs-de Leeuw-Glicksberg Decomposition.............................. 16.1 Compact Semigroups......................................................................... 16.2 Weakly Compact Operator Semigroups......................................... 16.3 The Jacobs-de Leeuw-Glicksberg Decomposition......................... 16.4 Cyclic Semigroups of Operators...................................................... Exercises........................................................................................................ 317 317 323 329 336 341 17 The Kronecker Factor and Systems with Discrete Spectrum .............................................................................................. 345 17.1 Semigroups of Markov Operators and the Kronecker
Factor........ 17.2 Dynamical Systems with Discrete Spectrum................................... 17.3 Disjointness of Weak Mixing and Discrete Spectrum..................... 17.4 Examples............................................................................................. Exercises........................................................................................................ 345 349 353 355 363 The Spectral Theorem and Dynamical Systems .................................. 18.1 The Spectral Theorem........................................................................ 18.2 Spectral Decompositions and the Maximal Spectral Type.............. 18.3 Discrete Measures and Eigenvalues.................................................. 18.4 Dynamical Systems............................................................................ Exercises ........................................................................................................ 367 367 376 381 386 398 18
Contents xviii 19 Topological Dynamics and Colorings........................................................... 19.1 The Stone-Čech Compactification.................................................... 19.2 The Semigroup Structure on ß5........................................................ 19.3 Topological Dynamics Revisited....................................................... 19.4 Hindman’s Theorem........................................................................... 19.5 From Coloring to Recurrence Results .............................................. 19.6 From Recurrence to Coloring Results ............................................. Exercises........................................................................................................ 20 Arithmetic Progressions and Ergodic Theory ......................................... 20.1 From Ergodic Theory to Arithmetic Progressions......................... 20.2 Back from Arithmetic Progressions to Ergodic Theory................. 20.3 The Host-Kra Theorem.................................................................... 20.4 Furstenberg’s Multiple Recurrence Theorem.................................. 20.5 The Furstenberg-Sárközy Theorem.................................................. Exercises........................................................................................................ 405 406 409 412 416 419 424 429 433 434 437 444 451 455 457 21.1 Weighted Ergodic Theorems............................................................. 21.2 Wiener-Wintner and Return Time Theorems
.................................. 21.3 Linear Sequences as Good Weights................................................... 21.4 Subsequential Ergodic Theorems...................................................... 21.5 Even More Ergodic Theorems.......................................................... Exercises........................................................................................................ 461 461 464 469 470 474 476 A Topology.............................................................................................................. 479 В Measure and Integration Theory.................................................................. 491 C Functional Analysis......................................................................................... D Operator Theory on Hilbert Spaces............................................................. 521 E The Riesz Representation Theorem............................................................. F Standard Probability Spaces.......................................................................... 557 21 More Ergodic Theorems .............................................................................. 507 543 G Theorems of Eberlein, Grothendieck, and Ellis........................................ 563 Bibliography.............................................................................................................. 577 Index............................................................................................................................ 613 Symbol
Index.......................................................................... 625
|
| any_adam_object | 1 |
| author | Eisner, Tatjana 1980- Farkas, Balint Haase, Markus 1970- Nagel, Rainer 1940- |
| author_GND | (DE-588)133328031 (DE-588)1060737418 (DE-588)141517956 |
| author_facet | Eisner, Tatjana 1980- Farkas, Balint Haase, Markus 1970- Nagel, Rainer 1940- |
| author_role | aut aut aut aut |
| author_sort | Eisner, Tatjana 1980- |
| author_variant | t e te b f bf m h mh r n rn |
| building | Verbundindex |
| bvnumber | BV043193663 |
| classification_rvk | SK 620 |
| ctrlnum | (OCoLC)934133519 (DE-599)GBV840599730 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV043193663 |
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| indexdate | 2024-07-10T07:20:12Z |
| institution | BVB |
| isbn | 9783319168975 |
| language | English |
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| publisher | Springer |
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| series | Graduate texts in mathematics |
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| spelling | Eisner, Tatjana 1980- Verfasser (DE-588)133328031 aut Operator theoretic aspects of ergodic theory Tanja Eisner ; Bálint Farkas ; Markus Haase ; Rainer Nagel Cham [u.a.] Springer 2015 XVIII, 628 S. Ill. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 272 Literaturverz. S. 577 - 611 Operatortheorie (DE-588)4075665-8 gnd rswk-swf Ergodentheorie (DE-588)4015246-7 gnd rswk-swf Ergodentheorie (DE-588)4015246-7 s Operatortheorie (DE-588)4075665-8 s DE-604 Farkas, Balint Verfasser aut Haase, Markus 1970- Verfasser (DE-588)1060737418 aut Nagel, Rainer 1940- Verfasser (DE-588)141517956 aut Erscheint auch als Online-Ausgabe 978-3-319-16898-2 Graduate texts in mathematics 272 (DE-604)BV000000067 272 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028617230&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Eisner, Tatjana 1980- Farkas, Balint Haase, Markus 1970- Nagel, Rainer 1940- Operator theoretic aspects of ergodic theory Graduate texts in mathematics Operatortheorie (DE-588)4075665-8 gnd Ergodentheorie (DE-588)4015246-7 gnd |
| subject_GND | (DE-588)4075665-8 (DE-588)4015246-7 |
| title | Operator theoretic aspects of ergodic theory |
| title_auth | Operator theoretic aspects of ergodic theory |
| title_exact_search | Operator theoretic aspects of ergodic theory |
| title_full | Operator theoretic aspects of ergodic theory Tanja Eisner ; Bálint Farkas ; Markus Haase ; Rainer Nagel |
| title_fullStr | Operator theoretic aspects of ergodic theory Tanja Eisner ; Bálint Farkas ; Markus Haase ; Rainer Nagel |
| title_full_unstemmed | Operator theoretic aspects of ergodic theory Tanja Eisner ; Bálint Farkas ; Markus Haase ; Rainer Nagel |
| title_short | Operator theoretic aspects of ergodic theory |
| title_sort | operator theoretic aspects of ergodic theory |
| topic | Operatortheorie (DE-588)4075665-8 gnd Ergodentheorie (DE-588)4015246-7 gnd |
| topic_facet | Operatortheorie Ergodentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028617230&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
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