Topics in symbolic dynamics and applications:
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Univ. Press
2000
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | London Mathematical Society lecture note series
279 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVI, 245 S. graph. Darst. |
| ISBN: | 0521796601 |
Internformat
MARC
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-009583094 | ||
Datensatz im Suchindex
| _version_ | 1804128853414641664 |
|---|---|
| adam_text | Contents
1 SEQUENCES OF LOW COMPLEXITY:
AUTOMATIC AND STURMIAN SEQUENCES 1
1.1 Introduction 1
1.2 Complexity Function 2
1.2.1 Definition 2
1.2.2 Frequencies and Measure Theoretic Entropy 4
1.2.3 Variational Principle 5
1.3 Symbolic Dynamical Systems 6
1.4 The Graph of Words 8
1.4.1 The Line Graph 10
1.4.2 Graph and Frequencies 11
1.5 Special factors 13
1.6 Sturmian sequences 14
1.6.1 A Particular Coding of Rotations 14
1.6.2 Frequencies of Factors of Sturmian Sequences 16
1.7 Automatic Sequences 17
1.7.1 Automata and Transcendence 18
1.7.2 Applications 19
1.7.3 The Multidimensional Case 21
1.7.4 Application to Diagonals 22
1.7.5 Transcendence of the Bracket Series 24
1.7.6 Complexity and Frequencies 26
1.8 Conclusion 28
1.8.1 Automaticity and Sturmian sequences 28
1.8.2 Sub affine Complexity 28
2 SUBSTITUTION SUBSHIFTS AND
BRATTELI DIAGRAMS 35
2.1 Subshifts 36
2.1.1 Notation: Words, Sequences, Morphisms 36
2.1.2 Subshifts 37
2.1.3 Minimal Systems 38
2.2 Substitutions 39
v
vi CONTENTS
2.2.1 Substitution Subshifts 39
2.2.2 Fixed Points 40
2.3 Unique Ergodicity 42
2.3.1 Perron Frobenius Theorem 42
2.3.2 Density of Letters 43
2.3.3 The Substitution ak on the Words of Length k 44
2.4 Structure of Substitution Subshifts 45
2.4.1 Structure of Substitution Dynamical Systems 45
2.5 Substitutions and Bratteli Diagrams 48
2.5.1 Bratteli Diagram Associated to a Substitution 48
2.5.2 The Vershik Map 50
2.5.3 An Isomorphism 51
2.5.4 The General Case 53
3 ALGEBRAIC ASPECTS OF SYMBOLIC DYNAMICS 57
3.1 Introduction 57
3.2 General Subshifts 58
3.2.1 Dynamical Systems 58
3.2.2 Full Shifts 58
3.2.3 Subshifts 59
3.2.4 Examples 59
3.3 Block Codes 60
3.3.1 Codes 60
3.3.2 Block Codes 60
3.3.3 Curtis Hedlund Lyndon 60
3.3.4 Higher Block Presentations 61
3.3.5 One Block Codes 61
3.4 Shifts of Finite Type 62
3.4.1 Vertex Shifts 62
3.4.2 Edge Shifts 63
3.4.3 Matrices 64
3.4.4 Markov Characterization 64
3.4.5 Applications 64
3.5 SFT like Subshifts 65
3.5.1 Sofic Shifts 65
3.5.2 Specification 66
3.5.3 Synchronized Systems 66
3.5.4 Coded Systems 66
3.5.5 The Progression 66
3.6 Minimal Subshifts 67
3.7 Matrix Invariants for SFTS 67
3.7.1 Nonnegative Matrices 68
3.7.2 Transitivity 69
CONTENTS vii
3.7.3 Mixing 69
3.7.4 Entropy 70
3.7.5 Periodic Points 70
3.7.6 Zeta Function 71
3.7.7 Isomorphism 72
3.7.8 Eventual Isomorphism 73
3.7.9 Flow Equivalence 74
3.7.10 Relations 75
3.8 Dimension Groups and Shift Equivalence 75
3.8.1 Dimension Groups 75
3.8.2 Dimension Modules 77
3.8.3 Shift Equivalence 79
3.8.4 Further Developments 81
3.9 Automorphisms and Classification of
SFTS 82
3.9.1 Automorphisms 82
3.9.2 Representations 83
3.10 The KRW Factorization Theorem 85
3.10.1 Statement of the Theorem 85
3.10.2 Nonsurjectivity of p 85
3.10.3 A Long Story 86
3.11 Classification 86
3.11.1 SE does not imply SSE: the Reducible Case 86
3.11.2 SE does not imply SSE: the Irreducible Case 87
4 DYNAMICS OF Zd ACTIONS ON MARKOV
SUBGROUPS 89
4.1 Introduction 89
4.2 One Dimensional Markov Subgroups 90
4.3 Decidability in One and Two Dimensions 98
4.4 Markov Subgroups of (Z/2Z)z2 104
4.5 Markov Subgroups Polynomial Rings 109
4.6 Conjugacy and Isomorphism in (Z/2Z)z2 117
4.7 General Zd Actions 118
5 ASYMPTOTIC LAWS FOR SYMBOLIC DYNAMICAL
SYSTEMS 123
5.1 Introduction 123
5.2 Preliminaries 124
5.2.1 Bernoulli Trials 124
5.2.2 Occurrence and Waiting Times 127
5.3 General Setup and Motivation 132
5.3.1 Return Maps and Expected Return Times 132
viii CONTENTS
5.3.2 Asymptotically Rare Events 134
5.3.3 Known Results and Motivation 135
5.4 Shifts of Finite Type and Equilibrium
States 143
5.4.1 Holder Potentials 144
5.4.2 Entropy and Pressure 145
5.4.3 Ruelle Perron Frobenius Operator 146
5.4.4 The Central Limit Theorem 147
5.4.5 Pianigiani Yorke Measure 149
5.5 Point Processes and Convergence in Law 153
5.5.1 Convergence of Point Processes 154
5.5.2 Entrance Times and Visiting Times 155
5.5.3 Final Remarks 160
5.5.4 Some Questions 161
6 ERGODIC THEORY AND DIOPHANTINE PROBLEMS 167
6.1 Introduction 167
6.2 Some Diophantine Problems Related to Polynomials and their
Connections with Combinatorics and Dynamics 169
6.3 Ramsey Theory and Topological
Dynamics 177
6.4 Density Ramsey Theory and Ergodic
Theory of Multiple Recurrence 183
6.5 Polynomial Ergodic Theorems and
Ramsey Theory 197
6.6 Appendix 200
7 NUMBER REPRESENTATION AND FINITE
AUTOMATA 207
7.1 Introduction 207
7.2 Words and Finite Automata 208
7.3 Standard Representations of Numbers 210
7.3.1 Representation of Integers 210
7.3.2 Representation of Real Numbers 211
7.3.3 fr Recognizable Sets of Integers 212
7.4 Beta Expansions 212
7.4.1 Definitions 212
7.4.2 The £ Shift 214
7.4.3 Classes of Numbers 216
7.4.4 Normalization in Base /? 218
7.5 [/ Representations 219
7.5.1 Definitions 219
7.5.2 The Set L(U) 220
CONTENTS ix
7.5.3 Normalization in the Linear Numeration System U . . 222
7.5.4 [/ Recognizable Sets of Integers 224
8 A NOTE ON THE TOPOLOGICAL CLASSIFICATION
OF LORENZ MAPS ON THE INTERVAL 229
8.1 Introduction 229
8.2 Statements of the Results 230
8.3 Proof of the Results 237
8.4 The General Case 241
|
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| discipline | Mathematik |
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| genre_facet | Konferenzschrift 1997 Temuco Kongress - Temuco <1997> - Symbolische Dynamik |
| id | DE-604.BV013999152 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:55:50Z |
| institution | BVB |
| isbn | 0521796601 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009583094 |
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| physical | XVI, 245 S. graph. Darst. |
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| series | London Mathematical Society lecture note series |
| series2 | London Mathematical Society lecture note series |
| spelling | Topics in symbolic dynamics and applications ed. by F. Blanchard ... 1. publ. Cambridge Univ. Press 2000 XVI, 245 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society lecture note series 279 Symbolische Dynamik (DE-588)4482797-0 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 1997 Temuco gnd-content Kongress - Temuco <1997> - Symbolische Dynamik Symbolische Dynamik (DE-588)4482797-0 s DE-604 Blanchard, François Sonstige oth London Mathematical Society lecture note series 279 (DE-604)BV000000130 279 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009583094&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Topics in symbolic dynamics and applications London Mathematical Society lecture note series Symbolische Dynamik (DE-588)4482797-0 gnd |
| subject_GND | (DE-588)4482797-0 (DE-588)1071861417 |
| title | Topics in symbolic dynamics and applications |
| title_auth | Topics in symbolic dynamics and applications |
| title_exact_search | Topics in symbolic dynamics and applications |
| title_full | Topics in symbolic dynamics and applications ed. by F. Blanchard ... |
| title_fullStr | Topics in symbolic dynamics and applications ed. by F. Blanchard ... |
| title_full_unstemmed | Topics in symbolic dynamics and applications ed. by F. Blanchard ... |
| title_short | Topics in symbolic dynamics and applications |
| title_sort | topics in symbolic dynamics and applications |
| topic | Symbolische Dynamik (DE-588)4482797-0 gnd |
| topic_facet | Symbolische Dynamik Konferenzschrift 1997 Temuco Kongress - Temuco <1997> - Symbolische Dynamik |
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