Symbolic dynamics is essential in the study of dynamical systems of various types and is connected to many other fields such as stochastic processes, ergodic theory, representation of numbers, information and coding, etc. This graduate text introduces symbolic dynamics from a perspective of topological dynamical systems and presents a vast variety of important examples.
After introducing symbolic and topological dynamics, the core of the book consists of discussions of various subshifts of positive entropy, of zero entropy, other non-shift minimal action on the Cantor set, and a study of the ergodic properties of these systems. The author presents recent developments such as spacing shifts, square-free shifts, density shifts, B-free shifts, Bratteli-Vershik systems, enumeration scales, amorphic complexity, and a modern and complete treatment of kneading theory. Later, he provides an overview of automata and linguistic complexity (Chomsky's hierarchy).
The necessary background for the book varies, but for most of it a solid knowledge of real analysis and linear algebra and first courses in probability and measure theory, metric spaces, number theory, topology, and set theory suffice. Most of the exercises have solutions in the back of the book.
Author(s): Henk Bruin
Series: Graduate Studies in Mathematics 228
Edition: 1
Publisher: American Mathematical Society
Year: 2022
Language: English
Pages: 460
Tags: Symbolic Dynamics, Subshifts, Entropy, Cantor Systems, Ergodic Theory, Linguistic Complexity
Cover
Title page
Contents
Preface
Chapter 1. First Examples and General Properties of Subshifts
1.1. Symbol Sequences and Subshifts
1.2. Word-Complexity
1.3. Transitive and Synchronized Subshifts
1.4. Sliding Block Codes
1.5. Word-Frequencies and Shift-Invariant Measures
1.6. Symbolic Itineraries
Chapter 2. Topological Dynamics
2.1. Basic Notions from Dynamical Systems
2.2. Transitive and Minimal Systems
2.3. Equicontinuous and Distal Systems
2.4. Topological Entropy
2.5. Mathematical Chaos
2.6. Transitivity and Topological Mixing
2.7. Shadowing and Specification
Chapter 3. Subshifts of Positive Entropy
3.1. Subshifts of Finite Type
3.2. Sofic Shifts
3.3. Coded Subshifts
3.4. Hereditary and Density Shifts
3.5. ?-Shifts and ?-Expansions
3.6. Unimodal Subshifts
3.7. Gap Shifts
3.8. Spacing Shifts
3.9. Power-Free Shifts
3.10. Dyck Shifts
Chapter 4. Subshifts of Zero Entropy
4.1. Linear Recurrence
4.2. Substitution Shifts
4.3. Sturmian Subshifts
4.4. Interval Exchange Transformations
4.5. Toeplitz Shifts
4.6. \cB-Free Shifts
4.7. Unimodal Restrictions to Critical Omega-Limit Sets
Chapter 5. Further Minimal Cantor Systems
5.1. Kakutani-Rokhlin Partitions
5.2. Cutting and Stacking
5.3. Enumeration Systems
5.4. Bratteli Diagrams and Vershik Maps
Chapter 6. Methods from Ergodic Theory
6.1. Ergodicity
6.2. Birkhoff’s Ergodic Theorem
6.3. Unique Ergodicity
6.4. Measure-Theoretic Entropy
6.5. Isomorphic Systems
6.6. Measures of Maximal Entropy
6.7. Mixing
6.8. Spectral Properties
6.9. Eigenvalues of Bratteli-Vershik Systems
Chapter 7. Automata and Linguistic Complexity
7.1. Automata
7.2. The Chomsky Hierarchy
7.3. Automatic Sequences and Cobham’s Theorems
Chapter 8. Miscellaneous Background Topics
8.1. Pisot and Salem Numbers
8.2. Continued Fractions
8.3. Uniformly Distributed Sequences
8.4. Diophantine Approximation
8.5. Density and Banach Density
8.6. The Perron-Frobenius Theorem
8.7. Countable Graphs and Matrices
Appendix. Solutions to Exercises
Bibliography
Index
Back Cover
