This book is the first self-contained exposition of the fascinating link between dynamical systems and dimension groups. The authors explore the rich interplay between topological properties of dynamical systems and the algebraic structures associated with them, with an emphasis on symbolic systems, particularly substitution systems. It is recommended for anybody with an interest in topological and symbolic dynamics, automata theory or combinatorics on words. Intended to serve as an introduction for graduate students and other newcomers to the field as well as a reference for established researchers, the book includes a thorough account of the background notions as well as detailed exposition – with full proofs – of the major results of the subject. A wealth of examples and exercises, with solutions, serve to build intuition, while the many open problems collected at the end provide jumping-off points for future research.

Introduction page 1

1 Topological Dynamical Systems 7

1.1 Recurrent and Minimal Dynamical Systems 7

1.2 More on Shift Spaces 17

1.3 Shifts of Finite Type 29

1.4 Substitution Shifts 30

1.5 Sturmian and Arnoux-Rauzy Shifts 51

1.6 Toeplitz Shifts 64

1.7 Exercises 66

1.8 Notes 75

2 Ordered Groups 82

2.1 Ordered Abelian Groups 82

2.2 States 87

2.3 Direct Limits 91

2.4 Dimension Groups 97

2.5 Stationary Systems 103

2.6 Exercises 105

2.7 Notes 108

3 Ordered Cohomology Groups 111

3.1 Coboundaries 112

3.2 Gottschalk and Hedlund Theorem 114

3.3 Ordered Group of a Dynamical System 118

3.4 Cylinder Functions 122

3.5 Ordered Group of a Recurrent Shift Space 123

3.6 Factor Maps and Conjugacy 127

3.7 Ordered Groups of Induced Systems 128

3.8 Invariant Borel Probability Measures 131

3.9 Invariant Measures and States 142

3.10 Exercises 146

3.11 Notes 153

4 Partitions in Towers 156

4.1 Partitions in Towers 156

4.2 Ordered Group Associated with a Partition 162

4.3 Ordered Groups of Sequences of Partitions 166

4.4 Dimension Groups and Return Words 170

4.5 Dimension Groups and Rauzy Graphs 178

4.6 Dimension Groups of Substitution Shifts 184

4.7 Exercises 196

4.8 Notes 198

5 Bratteli Diagrams 200

5.1 Bratteli Diagrams 200

5.2 Dynamics for Ordered Bratteli Diagrams 209

5.3 The Bratteli-Vershik Model Theorem 211

5.4 Kakutani Equivalence 220

5.5 Orbit Equivalence 221

5.6 Equivalences on Cantor Spaces 228

5.7 Entropy and Bratteli Diagrams 233

5.8 Exercises 236

5.9 Notes 241

6 Substitution Shifts and Generalizations 244

6.1 Odometers 244

6.2 Substitutio Shifts 252

6.3 Linearly Recurrent Shifts 268

6.4 S-adic Representations 277

6.5 Dimension Groups of Unimodular Shifts 289

6.6 Derivatives of Substitutive Sequences 291

6.7 Exercises 298

6.8 Notes 304

6.9 Exercises 307

7 Dendric Shifts 309

7.1 Dendric Shifts 309

7.2 Sturmian Shifts

7.3 Specular Shifts

7.4 Exercises 331

7.5 Notes 335

8 Interval Exchange Transformations 359

8.1 Interval Exchange Transformations 360

8.2 Branching Rauzy Induction 385

8.3 Interval Exchange over a Quadratic Field 395

8.4 Linear Involutions 398

8.5 Exercises 406

8.6 Notes 408

9 Bratteli Diagrams and C*-Algebras 411

9.1 Bratteli Diagrams 411

9.2 C*-Algebras 414

9.3 Approximately Finite Algebras 419

9.4 Exercises 432

9.5 Notes 433

A Solutions to Exercises 435

A.1 Chapter 1 435

A.2 Chapter 2 454

A.3 Chapter 3 459

A.4 Chapter4 468

A.5 Chapter5 473

A.6 Chapter6 482

A.7 Chapter7 493

A.8 Chapter 8 500

A.9 Chapter 9 504

B Useful Definitions and Results 507

B.1 Algebraic Number Theory 507

B.2 Groups, Graphs and Algebras 511

B.3 Linear Algebra 518

B.4 Topological, Metric and Normed Spaces 521

B.5 Measure and Integration 527

B.6 Topological Entropy 532

C Summary of Examples 534

D Equivalent Definitions of Sturmian Shifts 538

E Open Problems 540

References 544

Name Index 559

Index of Symbols 563

Subject Index 566

Dominique Perrin is Emeritus Professor in Mathematics and Computer Science at Université Gustave Eiffel. He is (co)author or editor of a number of books including Profinite Semigroups and Symbolic Dynamics, Codes and Automata, Infinite Words and Combinatorics on Words (under the pseudonym Lothaire). He is a member of Academia Europea.

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