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Topics in Symbolic Dynamics and Applications

edited by F. Blanchard, A. Maass, A. Nogueira
Format
Book
Published
Cambridge, UK ; New York : Cambridge University Press, 2000.
Language
English
Series
London Mathematical Society Lecture Note Series
ISBN
0521796601 (pbk.)
Contents
  • 1 Sequences of Low Complexity: Automatic and Sturmian Sequences 1
  • 1.2 Complexity Function 2
  • 1.2.2 Frequencies and Measure-Theoretic Entropy 4
  • 1.2.3 Variational Principle 5
  • 1.3 Symbolic Dynamical Systems 6
  • 1.4 Graph of Words 8
  • 1.4.1 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 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
  • 1.8.1 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
  • 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 Substitution [sigma subscript kappa] on the Words of Length [kappa] 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 Vershik Map 50
  • 2.5.3 An Isomorphism 51
  • 2.5.4 General Case 53
  • 3 Algebraic Aspects of Symbolic Dynamics 57
  • 3.2 General Subshifts 58
  • 3.2.1 Dynamical Systems 58
  • 3.2.2 Full Shifts 58
  • 3.2.3 Subshifts 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 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
  • 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 KRW Factorization Theorem 85
  • 3.10.1 Statement of the Theorem 85
  • 3.10.2 Nonsurjectivity of [rho] 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 Z[superscript d] Actions on Markov Subgroups 89
  • 4.2 One-Dimensional Markov Subgroups 90
  • 4.3 Decidability in One and Two-Dimensions 98
  • 4.4 Markov Subgroups of (Z/2Z)[superscript Z superscript 2] 104
  • 4.5 Markov Subgroups Polynomial Rings 109
  • 4.6 Conjugacy and Isomorphism in (Z/2Z)[superscript Z superscript 2] 117
  • 4.7 General Z[superscript d] Actions 118
  • 5 Asymptotic Laws for Symbolic Dynamical Systems 123
  • 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
  • 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 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
  • 6 Ergodic Theory and Diophantine Problems 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
  • 7 Number Representation and Finite Automata 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 b-Recognizable Sets of Integers 212
  • 7.4 Beta-Expansions 212
  • 7.4.2 [Beta]-Shift 214
  • 7.4.3 Classes of Numbers 216
  • 7.4.4 Normalization in Base [beta] 218
  • 7.5 U-Representations 219
  • 7.5.2 Set L(U) 220
  • 7.5.3 Normalization in the Linear Numeration System U 222
  • 7.5.4 U-Recognizable Sets of Integers 224
  • 8 A Note on the Topological Classification of Lorenz Maps on the Interval 229
  • 8.2 Statements of the Results 230
  • 8.3 Proof of the Results 237
  • 8.4 General Case 241.
Description
xvi, 245 p. : ill. ; 23 cm.
Series Statement
London Mathematical Society lecture note series ; 279
Technical Details
  • Access in Virgo Classic
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    g| 1 t| Sequences of Low Complexity: Automatic and Sturmian Sequences g| 1 -- g| 1.2 t| Complexity Function g| 2 -- g| 1.2.2 t| Frequencies and Measure-Theoretic Entropy g| 4 -- g| 1.2.3 t| Variational Principle g| 5 -- g| 1.3 t| Symbolic Dynamical Systems g| 6 -- g| 1.4 t| Graph of Words g| 8 -- g| 1.4.1 t| Line Graph g| 10 -- g| 1.4.2 t| Graph and Frequencies g| 11 -- g| 1.5 t| Special factors g| 13 -- g| 1.6 t| Sturmian sequences g| 14 -- g| 1.6.1 t| A Particular Coding of Rotations g| 14 -- g| 1.6.2 t| Frequencies of Factors of Sturmian Sequences g| 16 -- g| 1.7 t| Automatic Sequences g| 17 -- g| 1.7.1 t| Automata and Transcendence g| 18 -- g| 1.7.2 t| Applications g| 19 -- g| 1.7.3 t| Multidimensional Case g| 21 -- g| 1.7.4 t| Application to Diagonals g| 22 -- g| 1.7.5 t| Transcendence of the Bracket Series g| 24 -- g| 1.7.6 t| Complexity and Frequencies g| 26 -- g| 1.8.1 t| Automaticity and Sturmian sequences g| 28 -- g| 1.8.2 t| Sub-affine Complexity g| 28 -- g| 2 t| Substitution Subshifts and Bratteli Diagrams g| 35 -- g| 2.1 t| Subshifts g| 36 -- g| 2.1.1 t| Notation: Words, Sequences, Morphisms g| 36 -- g| 2.1.2 t| Subshifts g| 37 -- g| 2.1.3 t| Minimal Systems g| 38 -- g| 2.2 t| Substitutions g| 39 -- g| 2.2.1 t| Substitution Subshifts g| 39 -- g| 2.2.2 t| Fixed Points g| 40 -- g| 2.3 t| Unique Ergodicity g| 42 -- g| 2.3.1 t| Perron-Frobenius Theorem g| 42 -- g| 2.3.2 t| Density of Letters g| 43 -- g| 2.3.3 t| Substitution [sigma subscript kappa] on the Words of Length [kappa] g| 44 -- g| 2.4 t| Structure of Substitution Subshifts g| 45 -- g| 2.4.1 t| Structure of Substitution Dynamical Systems g| 45 -- g| 2.5 t| Substitutions and Bratteli Diagrams g| 48 -- g| 2.5.1 t| Bratteli Diagram Associated to a Substitution g| 48 -- g| 2.5.2 t| Vershik Map g| 50 -- g| 2.5.3 t| An Isomorphism g| 51 -- g| 2.5.4 t| General Case g| 53 -- g| 3 t| Algebraic Aspects of Symbolic Dynamics g| 57 -- g| 3.2 t| General Subshifts g| 58 -- g| 3.2.1 t| Dynamical Systems g| 58 -- g| 3.2.2 t| Full Shifts g| 58 -- g| 3.2.3 t| Subshifts g| 59 -- g| 3.3 t| Block Codes g| 60 -- g| 3.3.1 t| Codes g| 60 -- g| 3.3.2 t| Block Codes g| 60 -- g| 3.3.3 t| Curtis-Hedlund-Lyndon g| 60 -- g| 3.3.4 t| Higher Block Presentations g| 61 -- g| 3.3.5 t| One-Block Codes g| 61 -- g| 3.4 t| Shifts of Finite Type g| 62 -- g| 3.4.1 t| Vertex Shifts g| 62 -- g| 3.4.2 t| Edge Shifts g| 63 -- g| 3.4.3 t| Matrices g| 64 -- g| 3.4.4 t| Markov Characterization g| 64 -- g| 3.4.5 t| Applications g| 64 -- g| 3.5 t| SFT-like Subshifts g| 65 -- g| 3.5.1 t| Sofic Shifts g| 65 -- g| 3.5.2 t| Specification g| 66 -- g| 3.5.3 t| Synchronized Systems g| 66 -- g| 3.5.4 t| Coded Systems g| 66 -- g| 3.5.5 t| Progression g| 66 -- g| 3.6 t| Minimal Subshifts g| 67 -- g| 3.7 t| Matrix Invariants for SFTS g| 67 -- g| 3.7.1 t| Nonnegative Matrices g| 68 -- g| 3.7.2 t| Transitivity g| 69 -- g| 3.7.3 t| Mixing g| 69 -- g| 3.7.4 t| Entropy g| 70 -- g| 3.7.5 t| Periodic Points g| 70 -- g| 3.7.6 t| Zeta Function g| 71 -- g| 3.7.7 t| Isomorphism g| 72 -- g| 3.7.8 t| Eventual Isomorphism g| 73 -- g| 3.7.9 t| Flow Equivalence g| 74 -- g| 3.7.10 t| Relations g| 75 -- g| 3.8 t| Dimension Groups and Shift Equivalence g| 75 -- g| 3.8.1 t| Dimension Groups g| 75 -- g| 3.8.2 t| Dimension Modules g| 77 -- g| 3.8.3 t| Shift Equivalence g| 79 -- g| 3.8.4 t| Further Developments g| 81 -- g| 3.9 t| Automorphisms and Classification of SFTS g| 82 -- g| 3.9.1 t| Automorphisms g| 82 -- g| 3.9.2 t| Representations g| 83 -- g| 3.10 t| KRW Factorization Theorem g| 85 -- g| 3.10.1 t| Statement of the Theorem g| 85 -- g| 3.10.2 t| Nonsurjectivity of [rho] g| 85 -- g| 3.10.3 t| A Long Story g| 86 -- g| 3.11 t| Classification g| 86 -- g| 3.11.1 t| SE does not imply SSE: the Reducible Case g| 86 -- g| 3.11.2 t| SE does not imply SSE: the Irreducible Case g| 87 -- g| 4 t| Dynamics of Z[superscript d] Actions on Markov Subgroups g| 89 -- g| 4.2 t| One-Dimensional Markov Subgroups g| 90 -- g| 4.3 t| Decidability in One and Two-Dimensions g| 98 -- g| 4.4 t| Markov Subgroups of (Z/2Z)[superscript Z superscript 2] g| 104 -- g| 4.5 t| Markov Subgroups Polynomial Rings g| 109 -- g| 4.6 t| Conjugacy and Isomorphism in (Z/2Z)[superscript Z superscript 2] g| 117 -- g| 4.7 t| General Z[superscript d] Actions g| 118 -- g| 5 t| Asymptotic Laws for Symbolic Dynamical Systems g| 123 -- g| 5.2.1 t| Bernoulli Trials g| 124 -- g| 5.2.2 t| Occurrence and Waiting Times g| 127 -- g| 5.3 t| General Setup and Motivation g| 132 -- g| 5.3.1 t| Return Maps and Expected Return Times g| 132 -- g| 5.3.2 t| Asymptotically Rare Events g| 134 -- g| 5.3.3 t| Known Results and Motivation g| 135 -- g| 5.4 t| Shifts of Finite Type and Equilibrium States g| 143 -- g| 5.4.1 t| Holder Potentials g| 144 -- g| 5.4.2 t| Entropy and Pressure g| 145 -- g| 5.4.3 t| Ruelle-Perron-Frobenius Operator g| 146 -- g| 5.4.4 t| Central Limit Theorem g| 147 -- g| 5.4.5 t| Pianigiani-Yorke Measure g| 149 -- g| 5.5 t| Point Processes and Convergence in Law g| 153 -- g| 5.5.1 t| Convergence of Point Processes g| 154 -- g| 5.5.2 t| Entrance Times and Visiting Times g| 155 -- g| 6 t| Ergodic Theory and Diophantine Problems g| 167 -- g| 6.2 t| Some Diophantine Problems Related to Polynomials and their Connections with Combinatorics and Dynamics g| 169 -- g| 6.3 t| Ramsey Theory and Topological Dynamics g| 177 -- g| 6.4 t| Density Ramsey Theory and Ergodic Theory of Multiple Recurrence g| 183 -- g| 6.5 t| Polynomial Ergodic Theorems and Ramsey Theory g| 197 -- g| 7 t| Number Representation and Finite Automata g| 207 -- g| 7.2 t| Words and Finite Automata g| 208 -- g| 7.3 t| Standard Representations of Numbers g| 210 -- g| 7.3.1 t| Representation of Integers g| 210 -- g| 7.3.2 t| Representation of Real Numbers g| 211 -- g| 7.3.3 t| b-Recognizable Sets of Integers g| 212 -- g| 7.4 t| Beta-Expansions g| 212 -- g| 7.4.2 t| [Beta]-Shift g| 214 -- g| 7.4.3 t| Classes of Numbers g| 216 -- g| 7.4.4 t| Normalization in Base [beta] g| 218 -- g| 7.5 t| U-Representations g| 219 -- g| 7.5.2 t| Set L(U) g| 220 -- g| 7.5.3 t| Normalization in the Linear Numeration System U g| 222 -- g| 7.5.4 t| U-Recognizable Sets of Integers g| 224 -- g| 8 t| A Note on the Topological Classification of Lorenz Maps on the Interval g| 229 -- g| 8.2 t| Statements of the Results g| 230 -- g| 8.3 t| Proof of the Results g| 237 -- g| 8.4 t| General Case g| 241.
    596
      
      
    a| 8
    650
      
    0
    a| Symbolic dynamics.
    700
    1
      
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    700
    1
      
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    1
      
    a| Nogueira, A., d| 1950-
    999
      
      
    a| QA611.5 .T48 2000 w| LC i| X004479749 l| STACKS m| MATH t| BOOK
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