Dimension Groups and Dynamical Systems
Substitutions, Bratteli Diagrams and Cantor Systems

This is the first self-contained exposition of the connections between symbolic dynamical systems, dimension groups and Bratteli diagrams.

Fabien Durand (Author), Dominique Perrin (Author)

9781108838689, Cambridge University Press

Hardback, published 3 February 2022

760 pages
23.6 x 15.8 x 4.2 cm, 1.039 kg

'This substantial monograph comprises the first thorough and self-contained exposition of the theory of dimension groups in topological dynamics … It is a welcome addition to the dynamics literature in an area of much activity with links to several other disciplines, including operator algebras and computer science.' Thomas B. Ward, zbMATH

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
1. Topological dynamical systems
2. Ordered groups
3. Ordered cohomology groups
4. Partitions in towers
5. Bratteli diagrams
6. Substitution shifts and generalizations
7. Dendric shifts
8. Interval exchange transformations
9. Bratteli diagrams and C*-algebras
Appendix A. Solutions to exercises
Appendix B. Useful definitions and results
Appendix C. Summary of examples
Appendix D. Equivalent definitions of Sturmian shifts
Appendix E. Open problems
References
Name index
Index of symbols
Subject index.

Subject Areas: Dynamics & statics [PHDT], Topology [PBP], Calculus & mathematical analysis [PBK], Groups & group theory [PBG], Algebra [PBF]