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E-grāmata: Dimension Groups and Dynamical Systems: Substitutions, Bratteli Diagrams and Cantor Systems

(Université de Picardie Jules Verne, Amiens),
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Dimension Groups and Dynamical Systems: Substitutions, Bratteli Diagrams and Cantor SystemsPalielināt
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"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"--

Recenzijas

'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 ' an impressive self-contained expository presentation that explores the connections between properties of topological dynamical systems and their dimension groups. The book is a rich source of material on topological and symbolic dynamics and will be useful for both graduate students in mathematics or computer science and experienced researchers.' Olena Karpel, MathSciNet 'A valuable resource, especially for those interested in the dimension groups and Vershik maps as related to low complexity subshifts.' Mike Boyle, Bulletin of the American Mathematical Society

Papildus informācija

This is the first self-contained exposition of the connections between symbolic dynamical systems, dimension groups and Bratteli diagrams.
Introduction 1(6)
1 Topological Dynamical Systems
7(75)
1.1 Recurrent and Minimal Dynamical Systems
7(10)
1.2 More on Shift Spaces
17(12)
1.3 Shifts of Finite Type
29(1)
1.4 Substitution Shifts
30(21)
1.5 Sturmian and Arnoux-Rauzy Shifts
51(13)
1.6 Toeplitz Shifts
64(2)
1.7 Exercises
66(9)
1.8 Notes
75(7)
2 Ordered Groups
82(29)
2.1 Ordered Abelian Groups
82(5)
2.2 States
87(4)
2.3 Direct Limits
91(6)
2.4 Dimension Groups
97(6)
2.5 Stationary Systems
103(2)
2.6 Exercises
105(3)
2.7 Notes
108(3)
3 Ordered Cohomology Groups
111(45)
3.1 Coboundaries
112(2)
3.2 Gottschalk and Hedlund Theorem
114(4)
3.3 Ordered Group of a Dynamical System
118(4)
3.4 Cylinder Functions
122(1)
3.5 Ordered Group of a Recurrent Shift Space
123(4)
3.6 Factor Maps and Conjugacy
127(1)
3.7 Ordered Groups of induced Systems
128(3)
3.8 Invariant Borel Probability Measures
131(11)
3.9 Invariant Measures and States
142(4)
3.10 Exercises
146(7)
3.11 Notes
153(3)
4 Partitions in Towers
156(44)
4.1 Partitions in Towers
156(6)
4.2 Ordered Group Associated with a Partition
162(4)
4.3 Ordered Groups of Sequences of Partitions
166(4)
4.4 Dimension Groups and Return Words
170(8)
4.5 Dimension Groups and Rauzy Graphs
178(6)
4.6 Dimension Groups of Substitution Shifts
184(12)
4.7 Exercises
196(2)
4.8 Notes
198(2)
5 Bratteli Diagrams
200(44)
5.1 Bratteli Diagrams
200(9)
5.2 Dynamics for Ordered Bratteli Diagrams
209(2)
5.3 The Bratteli-Vershik Model Theorem
211(9)
5.4 Kakutani Equivalence
220(1)
5.5 Orbit Equivalence
221(7)
5.6 Equivalences on Cantor Spaces
228(5)
5.7 Entropy and Bratteli Diagrams
233(3)
5.8 Exercises
236(5)
5.9 Notes
241(3)
6 Substitution Shifts and Generalizations
244(65)
6.1 Odometers
244(8)
6.2 Substitution Shifts
252(16)
6.3 Linearly Recurrent Shifts
268(9)
6.4 5-adic Representations
277(12)
6.5 Dimension Groups of Unimodular Shifts
289(2)
6.6 Derivatives of Substitutive Sequences
291(7)
6.7 Exercises
298(6)
6.8 Notes
304(3)
6.9 Exercises
307(2)
7 Dendric Shifts
309(50)
7.1 Dendric Shifts
309(22)
7.2 Sturmian Shifts
331(4)
7.3 Specular Shifts
335(18)
7.4 Exercises
353(3)
7.5 Notes
356(3)
8 Interval Exchange Transformations
359(52)
8.1 Interval Exchange Transformations
360(25)
8.2 Branching Rauzy Induction
385(10)
8.3 Interval Exchange over a Quadratic Field
395(3)
8.4 Linear Involutions
398(8)
8.5 Exercises
406(2)
8.6 Notes
408(3)
9 Bratteli Diagrams and C*-Algebras
411(24)
9.1 Bratteli Diagrams
411(3)
9.2 C*-Algebras
414(5)
9.3 Approximately Finite Algebras
419(13)
9.4 Exercises
432(1)
9.5 Notes
433(2)
A Solutions to Exercises
435(72)
A.1
Chapter 1
435(19)
A.2
Chapter 2
454(5)
A.3
Chapter 3
459(9)
A.4
Chapter 4
468(5)
A.5
Chapter 5
473(9)
A.6
Chapter 6
482(11)
A.7
Chapter 7
493(7)
A.8
Chapter 8
500(4)
A.9
Chapter 9
504(3)
B Useful Definitions and Results
507(27)
B.1 Algebraic Number Theory
507(4)
B.2 Groups, Graphs and Algebras
511(7)
B.3 Linear Algebra
518(3)
B.4 Topological, Metric and Normed Spaces
521(6)
B.5 Measure and Integration
527(5)
B.6 Topological Entropy
532(2)
C Summary of Examples
534(4)
D Equivalent Definitions of Sturmian Shifts
538(2)
E Open Problems
540(4)
References 544(15)
Name Index 559(4)
Index of Symbols 563(3)
Subject Index 566
Fabien Durand is Full Professor in Mathematics at Université de Picardie Jules Verne. His interests include topological dynamical systems and the relations with theoretical computer science. He is currently the president of the Société Mathématique de France. 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.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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