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9789810223984

Grammatical Complexity and One-Dimensional Dynamical Systems

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  • ISBN13:

    9789810223984

  • ISBN10:

    9810223986

  • Format: Hardcover
  • Copyright: 1997-01-01
  • Publisher: World Scientific Pub Co Inc
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Table of Contents

Preface v
Part I: Strings and Languages 1(36)
1 Free Monoids
3(10)
1.1 Free Monoids and Strings
3(5)
1.1.1 Definition of Monoids
3(2)
1.1.2 Free Monoids
5(1)
1.1.3 Primitive Strings
6(1)
1.1.4 Cyclic Shifts of Strings
7(1)
1.1.5 Other Operators and Notations
7(1)
1.2 An Ordered Free Monoid
8(5)
1.2.1 An Order Relation on the Monoid {0,1}*
8(1)
1.2.2 Maximal Strings
8(1)
1.2.3 Maximal and Primitive Strings
9(4)
2 Dynamical Languages
13(24)
2.1 Definition of Dynamical Languages
14(1)
2.2 Distinct Excluded Blocks
15(6)
2.2.1 Definition and Properties
15(2)
2.2.2 L and L" in Chomsky Hierarchy
17(2)
2.2.3 A Natural Equivalence Relation
19(2)
2.3 Symbolic Flows
21(5)
2.3.1 Symbolic Flows and Dynamical Languages
21(2)
2.3.2 Subshifts of Finite Type
23(2)
2.3.3 Sofic Systems
25(1)
2.4 Graphs and Dynamical Languages
26(7)
2.4.1 Graphs and Shannon-Graphs
27(2)
2.4.2 Transitive Languages
29(4)
2.5 Topological Entropy
33(4)
Part II: Grammatical Complexity of Unimodal Maps 37(142)
3 Languages of Unimodal Maps
39(24)
3.1 Unimodal Maps
39(4)
3.1.1 Definition and Terminology
39(2)
3.1.2 Logistic Map and Tent Map
41(1)
3.1.3 Simple Models with Complicated Dynamics
42(1)
3.2 Symbolic Dynamics
43(7)
3.2.1 A Brief History
43(1)
3.2.2 Itinerary and Kneading Sequence
44(1)
3.2.3 An Order Relation of Sequences
45(2)
3.2.4 Conditions of being Itinerary
47(3)
3.3 Definition of Languages
50(5)
3.3.1 Admissible Sequences
50(1)
3.3.2 The Language L(KS)
51(1)
3.3.3 Each Word is "Real"
51(3)
3.3.4 L(KS) are Dynamical Languages
54(1)
3.4 Some Facts of Strings for a Given Kneading Sequence
55(3)
3.4.1 Prefix-Suffixes with respect to Kneading Sequence
55(1)
3.4.2 Conditions for x E L(KS)
56(1)
3.4.3 L(KS) when KS contains c
57(1)
3.4.4 Two Lemmas about Dual Strings
57(1)
3.5 Periodic Sequences and Periodic Orbits
58(5)
3.5.1 Periodic Sequences which are Itineraries
58(2)
3.5.2 Complexity of Windows
60(3)
4 Regular Languages of Unimodal Maps
63(34)
4.1 Some Examples
63(5)
4.1.1 The Surjective Unimodal Map
63(3)
4.1.2 More Examples
66(2)
4.2 Decide Regularity from Kneading Sequence
68(9)
4.2.1 Calculations of Equivalence Classes
68(2)
4.2.2 The Role of Prefixes of Kneading Sequence
70(1)
4.2.3 The Necessary Part of Theorem 4.2.1
70(3)
4.2.4 The Maximal Path in minDFA
73(2)
4.2.5 The Sufficient Part of Theorem 4.2.1
75(2)
4.3 Minimum States DFA for Periodic Kneading Sequence
77(6)
4.3.1 Calculation of the Index of RL
78(2)
4.3.2 Construction of minDFA for KS = x
80(1)
4.3.3 Some Examples
81(2)
4.4 Minimum States DFA for Eventually Periodic Kneading Sequence
83(8)
4.4.1 Some Lemmas
83(1)
4.4.2 Calculation of the Index of RL
84(3)
4.4.3 Some Examples
87(1)
4.4.4 Construction of minDFA for KS = p
88(3)
4.5 Composition Laws and Self-Similarity
91(6)
4.5.1 The *-Composition Law
91(1)
4.5.2 Self-Similarity
92(2)
4.5.3 The Generalized Composition Laws
94(3)
5 A General Discussion of Kneading Sequences
97(14)
5.1 Maximal Primitive Prefixes of Kneading Sequences
97(6)
5.1.1 Existence of Special Prefixes of Kneading Sequences
98(1)
5.1.2 Odd Maximal Primitive Prefixes of Kneading Sequences
99(1)
5.1.3 Kneading Maps
100(3)
5.2 Infinite Automata of Unimodal Maps
103(3)
5.2.1 Infinite Automata
103(1)
5.2.2 Some Examples
104(2)
5.3 Density of Periodic Kneading Sequences
106(2)
5.3.1 Regard Kneading Sequence as a Limit
106(1)
5.3.2 Density of Periodic Kneading Sequences
107(1)
5.4 Existence of Uncomputable Complexity
108(3)
5.4.1 Uncountable Infinity of Kneading Sequences
108(1)
5.4.2 Some Lemmas
109(1)
5.4.3 Proof of Propositions 5.4.1 and 5.4.2
110(1)
6 Non-Regular Languages of Unimodal Maps
111(34)
6.1 The Language of Feigenbaum Attractor
111(9)
6.1.1 Renormalization and Kneading Sequence t of Feigenbaum Attractor
112(4)
6.1.2 Some Properties of t
116(2)
6.1.3 The Non-Regularity of L(t)
118(1)
6.1.4 The Thue-Morse Sequence
118(2)
6.2 Grammatical Level of Feigenbaum Attractor
120(6)
6.2.1 Structure of Language L = L(t)
120(2)
6.2.2 L is not a Context-Free Language
122(1)
6.2.3 L is a Context-Sensitive Language
123(3)
6.3 The Approach of Fibonacci Sequences
126(10)
6.3.1 Fibonacci Sequences and Cyclic Shifts
126(3)
6.3.2 Calculation of Cyclic Numbers
129(5)
6.3.3 Proof of Theorem 6.3.13
134(2)
6.4 Homomorphisms on Free Submonoids
136(9)
6.4.1 Two Special Kinds of Homomorphisms
136(3)
6.4.2 A More General Route
139(6)
7 DEB of Unimodal Maps
145(16)
7.1 A General Discussion
145(2)
7.1.1 Criterion of being DEB
145(1)
7.1.2 Finding the Next DEB
146(1)
7.2 Structure of Regular L"
147(4)
7.2.1 Finite Complement Languages L(KS)
147(1)
7.2.2 Semilinear Structure of Regular L"
148(1)
7.2.3 Calculation of Regular L"
149(2)
7.3 Complexity Levels of L and L"
151(3)
7.4 DEB of Non-Regular Languages
154(7)
7.4.1 DEB of Feigenbaum Attractor
154(1)
7.4.2 DEB of Even Fibonacci Languages
155(1)
7.4.3 DEB of Odd Fibonacci Languages
156(3)
7.4.4 Distribution of DEB
159(2)
8 Topological Entropy of Unimodal Maps
161(18)
8.1 Introduction
161(4)
8.1.1 Growth Number and Entropy
161(1)
8.1.2 The Sequence {Sn}
162(1)
8.1.3 Kneading Sequences and Entropy
163(1)
8.1.4 Some Examples
163(2)
8.2 General Recursive Relations of {Sn}
165(5)
8.2.1 The Main Theorem
166(2)
8.2.2 Difference Equation for Periodic KS
168(1)
8.2.3 Difference Equation for Eventually Periodic KS
169(1)
8.3 Kneading Invariant
170(2)
8.4 Applications
172(7)
8.4.1 A Theorem of Bowen and Frank
172(2)
8.4.2 Unimodal Maps with Positive Entropy
174(1)
8.4.3 Periodic Orbits with Odd Primitive Period
175(1)
8.4.4 Constant Entropy in Window
176(3)
Part III: Grammatical Complexity of Circle Homeomorphisms 179(48)
9 Languages of Circle Homeomorphisms
181(16)
9.1 Definition of Languages
182(3)
9.2 Rotation Number and Entropy
185(3)
9.2.1 Rotation Number of Circle Homeomorphisms
185(2)
9.2.2 Topological Entropy of Circle Homeomorphisms
187(1)
9.3 Nested Block Structure
188(9)
9.3.1 Existence of Nested Block Structure
188(3)
9.3.2 Farey Tree and Continued Fraction
191(3)
9.3.3 Relation of Nested Block Structure with Farey Address
194(3)
10 Complexity Levels of Circle Homeomorphisms
197(12)
10.1 Regular Languages for Rational Rotation Numbers
197(4)
10.1.1 Properties of Basic Word
198(1)
10.1.2 Proof of Theorem 10.1.1
199(2)
10.2 Non-Regular Languages for Irrational Rotation Numbers
201(4)
10.3 The Case of Golden Mean Ratio
205(4)
11 Automata of Circle Homeomorphisms
209(18)
11.1 Finite Automata of Circle Homeomorphisms
209(4)
11.2 Infinite Automata of Circle Homeomorphisms
213(2)
11.3 Universal Structure of Automata of Circle Homeomorphisms
215(12)
11.3.1 Main Results
216(1)
11.3.2 Strings h1...hk(1) and h1...hk(0)
217(1)
11.3.3 Square Prefixes of K+
218(1)
11.3.4 Calculation of {xn}
219(2)
11.3.5 Calculation of Prefixes of the First Kind
221(1)
11.3.6 Proof of Theorem 11.3.2
222(2)
11.3.7 Proof of Theorem 11.3.3
224(3)
Appendices 227(32)
A Finite Automata and Regular Languages 227(14)
A.1 Finite Automata 227(4)
A.1.1 Deterministic Finite Automata 227(2)
A.1.2 Extensions of DFA 229(1)
A.1.3 Transition Diagrams of FA 230(1)
A.2 Grammars and Expressions 231(2)
A.2.1 Right-Linear Grammars 231(1)
A.2.2 Regular Expressions 232(1)
A.3 Minimum States DFA 233(5)
A.3.1 A Natural Equivalence Relation RL 234(1)
A.3.2 The Role of RL 234(1)
A.3.3 The Myhill-Nerode Theorem 235(2)
A.3.4 Applications 237(1)
A.4 Properties of Regular Languages 238(3)
A.4.1 Pumping Lemma of Regular Languages 239(1)
A.4.2 Closure Properties 239(2)
B Non-Regular Languages 241(12)
B.1 Turing Machines 241(6)
B.1.1 Definitions 241(1)
B.1.2 Recursively Enumerable Languages 242(2)
B.1.3 Recursive Languages 244(1)
B.1.4 Linear Bounded Automata 245(1)
B.1.5 Pushdown Automata 246(1)
B.2 Languages Generated by Grammars 247(6)
B.2.1 The Chomsky Hierarchy 247(1)
B.2.2 Context-Free Languages 248(1)
B.2.3 Pumping Lemma of Context-Free Languages 249(1)
B.2.4 The Ogden Lemma 250(1)
B.2.5 Two Theorems related CFL and RGL 250(1)
B.2.6 Context-Sensitive Languages 251(2)
C L Systems and Languages 253(6)
C.1 Definitions of Some L Systems and Languages 253(4)
C.1.1 DOL Languages 253(2)
C.1.2 OL Languages 255(1)
C.1.3 TOL Languages 256(1)
C.1.4 ETOL Languages 256(1)
C.2 Relations with Chomsky Hierarchy 257(2)
C.2.1 Languages between CFL and CSL 257(1)
C.2.2 Full Abstract Family of Languages 257(2)
References 259(10)
Symbols Index 269(2)
Subject Index 271

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