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El. knyga: Topics in Symbolic Dynamics and Applications

Edited by (Institut de France, Paris), Edited by (Universidade Federal do Rio de Janeiro), Edited by (Universidad de Chile)
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The eight lectures given at the CIMPS-UNESCO summer school in Temuca, Chile, in January 1997 for graduate and advanced undergraduate students studying dynamic systems. Symbolic dynamics is now most often used for coding in ergodic information theory. The course is not designed as a textbook, but as a sampling of what is going on in the field. It is not indexed. Annotation c. Book News, Inc., Portland, OR (booknews.com)

This book is devoted to recent developments in symbolic dynamics, and it comprises eight chapters. The first two are concerned with the study of symbolic sequences of "low complexity," the following two introduce "high complexity" systems. Chapter five presents results on asymptotic laws for the random times of occurrence of rare events. Chapter six deals with diophantine problems and combinatorial Ramsey theory. Chapter seven looks at the dynamics of symbolic systems arising from numeration systems, and chapter eight gives a complete description of the symbolic dynamics of Lorenz maps.

This book is devoted to recent developments in symbolic dynamics.

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This book, first published in 2000, is devoted to recent developments in symbolic dynamics.
Sequences of Low Complexity: Automatic and Sturmian Sequences
1(34)
Introduction
1(1)
Complexity Function
2(4)
Definition
2(2)
Frequencies and Measure-Theoretic Entropy
4(1)
Variational Principle
5(1)
Symbolic Dynamical Systems
6(2)
The Graph of Words
8(5)
The Line Graph
10(1)
Graph and Frequencies
11(2)
Special factors
13(1)
Sturmian sequences
14(3)
A Particular Coding of Rotations
14(2)
Frequencies of Factors of Sturmian Sequences
16(1)
Automatic Sequences
17(11)
Automata and Transcendence
18(1)
Applications
19(2)
The Multidimensional Case
21(1)
Application to Diagonals
22(2)
Transcendence of the Bracket Series
24(2)
Complexity and Frequencies
26(2)
Conclusion
28(7)
Automaticity and Sturmian sequences
28(1)
Sub-affine Complexity
28(7)
Substitution Subshifts and Bratteli Diagrams
35(22)
Subshifts
36(3)
Notation: Words, Sequences, Morphisms
36(1)
Subshifts
37(1)
Minimal Systems
38(1)
Substitutions
39(3)
Substitution Subshifts
39(1)
Fixed Points
40(2)
Unique Ergodicity
42(3)
Perron-Frobenius Theorem
42(1)
Density of Letters
43(1)
The Substitution σk on the Words of Length k
44(1)
Structure of Substitution Subshifts
45(3)
Structure of Substitution Dynamical Systems
45(3)
Substitutions and Bratteli Diagrams
48(9)
Bratteli Diagram Associated to a Substitution
48(2)
The Vershik Map
50(1)
An Isomorphism
51(2)
The General Case
53(4)
Algebraic Aspects of Symbolic Dynamics
57(32)
Introduction
57(1)
General Subshifts
58(2)
Dynamical Systems
58(1)
Full Shifts
58(1)
Subshifts
59(1)
Examples
59(1)
Block Codes
60(2)
Codes
60(1)
Block Codes
60(1)
Curtis-Hedlund-Lyndon
60(1)
Higher Block Presentations
61(1)
One-Block Codes
61(1)
Shifts of Finite Type
62(3)
Vertex Shifts
62(1)
Edge Shifts
63(1)
Matrices
64(1)
Markov Characterization
64(1)
Applications
64(1)
SFT-like Subshifts
65(2)
Sofic Shifts
65(1)
Specification
66(1)
Synchronized Systems
66(1)
Coded Systems
66(1)
The Progression
66(1)
Minimal Subshifts
67(1)
Matrix Invariants for SFTS
67(8)
Nonnegative Matrices
68(1)
Transitivity
69(1)
Mixing
69(1)
Entropy
70(1)
Periodic Points
70(1)
Zeta Function
71(1)
Isomorphism
72(1)
Eventual Isomorphism
73(1)
Flow Equivalence
74(1)
Relations
75(1)
Dimension Groups and Shift Equivalence
75(7)
Dimension Groups
75(2)
Dimension Modules
77(2)
Shift Equivalence
79(2)
Further Developments
81(1)
Automorphisms and Classification of SFTS
82(3)
Automorphisms
82(1)
Representations
83(2)
The KRW Factorization Theorem
85(1)
Statement of the Theorem
85(1)
Nonsurjectivity of p
85(1)
A Long Story
86(1)
Classification
86(3)
SE does not imply SSE: the Reducible Case
86(1)
SE does not imply SSE: the Irreducible Case
87(2)
Dynamics of Zd Actions on Markov Subgroups
89(34)
Introduction
89(1)
One-Dimensional Markov Subgroups
90(8)
Decidability in One and Two-Dimensions
98(6)
Markov Subgroups of (Z/2Z)Z
104(5)
Markov Subgroups Polynomial Rings
109(8)
Conjugacy and Isomorphism in (Z/2Z)Z
117(1)
General Zd Actions
118(5)
Asymptotic Laws for Symbolic Dynamical Systems
123(44)
Introduction
123(1)
Preliminaries
124(8)
Bernoulli Trials
124(3)
Occurrence and Waiting Times
127(5)
General Setup and Motivation
132(11)
Return Maps and Expected Return Times
132(2)
Asymptotically Rare Events
134(1)
Known Results and Motivation
135(8)
Shifts of Finite Type and Equilibrium States
143(10)
Holder Potentials
144(1)
Entropy and Pressure
145(1)
Ruelle-Perron-Frobenius Operator
146(1)
The Central Limit Theorem
147(2)
Pianigiani-Yorke Measure
149(4)
Point Processes and Convergence in Law
153(14)
Convergence of Point Processes
154(1)
Entrance Times and Visiting Times
155(5)
Final Remarks
160(1)
Some Questions
161(6)
Ergodic Theory and Diophantine Problems
167(40)
Introduction
167(2)
Some Diophantine Problems Related to Polynomials and their Connections with Combinatorics and Dynamics
169(8)
Ramsey Theory and Topological Dynamics
177(6)
Density Ramsey Theory and Ergodic Theory of Multiple Recurrence
183(14)
Polynomial Ergodic Theorems and Ramsey Theory
197(3)
Appendix
200(7)
Number Representation and Finite Automata
207(22)
Introduction
207(1)
Words and Finite Automata
208(2)
Standard Representations of Numbers
210(2)
Representation of Integers
210(1)
Representation of Real Numbers
211(1)
b-Recognizable Sets of Integers
212(1)
Beta-Expansions
212(7)
Definitions
212(2)
The β-Shift
214(2)
Classes of Numbers
216(2)
Normalization in Base β
218(1)
U-Representations
219(10)
Definitions
219(1)
The Set L(U)
220(2)
Normalization in the Linear Numeration System U
222(2)
U-Recognizable Sets of Integers
224(5)
A Note on the Topological Classification of Lorenz Maps on the Interval
229
Introduction
229
Statements of the Results
230
Proof of the Results
237
The General Case
241
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