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Topological and Ergodic Theory of Symbolic Dynamics by Henk Bruin, ISBN-13: 978-1470469849

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Topological and Ergodic Theory of Symbolic Dynamics by Henk Bruin, ISBN-13: 978-1470469849

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  • Publisher: ‎ American Mathematical Society (January 20, 2023)
  • Language: ‎ English
  • 460 pages
  • ISBN-10: ‎ 1470469847
  • ISBN-13: ‎ 978-1470469849

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, $mathcal{B}$-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.

Table of Contents:

Preface ix
Chapter 1. First Examples and General Properties of Subshifts 1
§1.1. Symbol Sequences and Subshifts 1
§1.2. Word-Complexity 5
§1.3. Transitive and Synchronized Subshifts 9
§1.4. Sliding Block Codes 10
§1.5. Word-Frequencies and Shift-Invariant Measures 12
§1.6. Symbolic Itineraries 14
Chapter 2. Topological Dynamics 19
§2.1. Basic Notions from Dynamical Systems 19
§2.2. Transitive and Minimal Systems 23
§2.3. Equicontinuous and Distal Systems 28
§2.4. Topological Entropy 36
§2.5. Mathematical Chaos 40
§2.6. Transitivity and Topological Mixing 44
§2.7. Shadowing and Specification 47
Chapter 3. Subshifts of Positive Entropy 51
§3.1. Subshifts of Finite Type 51
§3.2. Sofic Shifts 61
§3.3. Coded Subshifts 65
§3.4. Hereditary and Density Shifts 71
§3.5. β-Shifts and β-Expansions 77
§3.6. Unimodal Subshifts 88
§3.7. Gap Shifts 117
§3.8. Spacing Shifts 120
§3.9. Power-Free Shifts 122
§3.10. Dyck Shifts 128
Chapter 4. Subshifts of Zero Entropy 133
§4.1. Linear Recurrence 133
§4.2. Substitution Shifts 135
§4.3. Sturmian Subshifts 162
§4.4. Interval Exchange Transformations 180
§4.5. Toeplitz Shifts 185
§4.6. B-Free Shifts 195
§4.7. Unimodal Restrictions to Critical Omega-Limit Sets 203
Chapter 5. Further Minimal Cantor Systems 217
§5.1. Kakutani-Rokhlin Partitions 217
§5.2. Cutting and Stacking 220
§5.3. Enumeration Systems 225
§5.4. Bratteli Diagrams and Vershik Maps 233
Chapter 6. Methods from Ergodic Theory 257
§6.1. Ergodicity 259
§6.2. Birkhoff’s Ergodic Theorem 260
§6.3. Unique Ergodicity 262
§6.4. Measure-Theoretic Entropy 282
§6.5. Isomorphic Systems 284
§6.6. Measures of Maximal Entropy 287
§6.7. Mixing 295
§6.8. Spectral Properties 309
§6.9. Eigenvalues of Bratteli-Vershik Systems 325
Chapter 7. Automata and Linguistic Complexity 341
§7.1. Automata 341
§7.2. The Chomsky Hierarchy 345
§7.3. Automatic Sequences and Cobham’s Theorems 357
Chapter 8. Miscellaneous Background Topics 367
§8.1. Pisot and Salem Numbers 367
§8.2. Continued Fractions 376
§8.3. Uniformly Distributed Sequences 385
§8.4. Diophantine Approximation 391
§8.5. Density and Banach Density 395
§8.6. The Perron-Frobenius Theorem 398
§8.7. Countable Graphs and Matrices 401
Appendix. Solutions to Exercises 413
Bibliography 423
Index 451

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