This book describes the relation between profinite semigroups and symbolic dynamics. Profinite semigroups are topological semigroups which are compact and residually finite. In particular, free profinite semigroups can be seen as the completion of free semigroups with respect to the profinite metric. In this metric, two words are close if one needs a morphism on a large finite monoid to distinguish them. The main focus is on a natural correspondence between minimal shift spaces (closed shift-invariant sets of two-sided infinite words) and maximal J-classes (certain subsets of free profinite semigroups). This correspondence sheds light on many aspects of both profinite semigroups and symbolic dynamics. For example, the return words to a given word in a shift space can be related to the generators of the group of the corresponding J-class. The book is aimed at researchers and graduate students in mathematics or theoretical computer science.
Author(s): Jorge Almeida, Alfredo Costa, Revekka Kyriakoglou
Series: Lecture Notes in Mathematics, 2274
Publisher: Springer
Year: 2020
Language: English
Pages: 290
City: Cham
Contents
1 Introduction
2 Prelude: Profinite Integers
2.1 Introduction
2.2 Profinite Integers
2.3 Profinite Natural Integers
2.4 Zero Set of a Recognizable Series
2.5 Odometers
2.6 Exercises
2.6.1 Section 2.2
2.6.2 Section 2.3
2.6.3 Section 2.4
2.7 Solutions
2.7.1 Section 2.2
2.7.2 Section 2.3
2.7.3 Section 2.4
2.8 Notes
3 Profinite Groups and Semigroups
3.1 Introduction
3.2 Topological and Metric Spaces
3.2.1 Topological Spaces
Definition and First Examples
Nets
Continuity
3.2.2 Metric Spaces
3.2.3 Compact Spaces
3.3 Topological Semigroups
3.3.1 Semigroups and Monoids
3.3.2 Interplay Between Algebra and Topology
3.3.3 Generating Sets
3.4 Topological Groups
3.5 Quotients of Compact Semigroups
3.6 Ideals and Green's Relations
3.6.1 Green's Relations
3.6.2 Stable Semigroups
3.6.3 The Schützenberger Group
3.7 The ω-Power
3.8 Projective Limits
3.9 Profinite Semigroups: Definition and Characterizations
3.10 Profinite Groups and Profinite Monoids
3.11 The Profinite Distance
3.12 Profinite Monoids of Continuous Endomorphisms
3.13 Exercises
3.13.1 Section 3.2
3.13.2 Section 3.3
3.13.3 Section 3.5
3.13.4 Section 3.6
3.13.5 Section 3.7
3.13.6 Section 3.8
3.13.7 Section 3.9
3.13.8 Section 3.12
3.14 Solutions
3.14.1 Section 3.2
3.14.2 Section 3.3
3.14.3 Section 3.5
3.14.4 Section 3.6
3.14.5 Section 3.7
3.14.6 Section 3.8
3.14.7 Section 3.9
3.14.8 Section 3.12
3.15 Notes
4 Free Profinite Monoids, Semigroups and Groups
4.1 Introduction
4.2 Free Monoids and Semigroups
4.3 Free Groups
4.4 Free Profinite Monoids and Semigroups
4.5 Pseudowords as Operations
4.6 Free Profinite Groups
4.7 Presentations of Profinite Semigroups
4.8 Profinite Codes
4.9 Relatively Free Profinite Monoids and Semigroups
4.10 Exercises
4.10.1 Section 4.2
4.10.2 Section 4.3
4.10.3 Section 4.4
4.10.4 Section 4.5
4.10.5 Section 4.6
4.10.6 Section 4.7
4.10.7 Section 4.8
4.10.8 Section 4.9
4.11 Solutions
4.11.1 Section 4.2
4.11.2 Section 4.3
4.11.3 Section 4.4
4.11.4 Section 4.5
4.11.5 Section 4.6
4.11.6 Section 4.7
4.11.7 Section 4.8
4.11.8 Section 4.9
4.12 Notes
5 Shift Spaces
5.1 Introduction
5.2 Factorial Sets
5.3 Shift Spaces
5.4 Block Maps and Conjugacy
5.5 Substitutive Shift Spaces
5.5.1 Primitive Substitutions
5.5.2 Matrix of a Substitution
5.5.3 Recognizable Substitutions
5.6 The Topological Closure of a Uniformly Recurrent Set
5.6.1 Uniformly Recurrent Pseudowords
5.6.2 The J-Class of a Uniformly Recurrent Set
5.7 Generalization to Recurrent Sets
5.8 Exercises
5.8.1 Section 5.2
5.8.2 Section 5.3
5.8.3 Section 5.5
5.8.4 Section 5.6
5.8.5 Section 5.7
5.9 Solutions
5.9.1 Section 5.2
5.9.2 Section 5.3
5.9.3 Section 5.5
5.9.4 Section 5.6
5.9.5 Section 5.7
5.10 Notes
6 Sturmian Sets and Tree Sets
6.1 Introduction
6.2 Return Words
6.2.1 Left and Right Return Words
6.3 Neutral Sets
6.4 Episturmian Words
6.5 Tree Sets
6.6 Sequences of Return Sets
6.7 Limit Return Sets
6.8 Exercises
6.8.1 Section 6.2
6.8.2 Section 6.3
6.8.3 Section 6.4
6.8.4 Section 6.5
6.9 Solutions
6.9.1 Section 6.2
6.9.2 Section 6.3
6.9.3 Section 6.4
6.9.4 Section 6.5
6.10 Notes
7 The Schützenberger Group of a Minimal Set
7.1 Introduction
7.2 Invariance of the Schützenberger Group
7.3 A Sufficient Condition for Freeness
7.4 Groups of Substitutive Shifts
7.4.1 Proper Substitutions
7.5 Exercises
7.5.1 Section 7.2
7.5.2 Section 7.3
7.5.3 Section 7.4
7.6 Solutions
7.6.1 Section 7.2
7.6.2 Section 7.3
7.6.3 Section 7.4
7.7 Notes
8 Groups of Bifix Codes
8.1 Introduction
8.2 Prefix Codes in Factorial Sets
8.2.1 Prefix Codes
8.2.2 Maximal Prefix Codes
8.2.3 Minimal Automata of Prefix Codes
8.3 Bifix Codes in Recurrent Sets
8.3.1 Group Codes
8.3.2 Parses
8.3.3 Complete Bifix Codes
8.4 Bifix Codes in Tree Sets
8.4.1 Cardinality Theorem
8.4.2 The Finite Index Basis Theorem
8.5 The Syntactic Monoid of a Recognizable Bifix Code
8.5.1 The F-Minimum J-Class
8.5.2 The F-Group as a Permutation Group
8.5.3 The Minimal Automaton of a Recognizable Bifix Code
8.6 The Charged Code Theorem
8.6.1 Charged Codes
8.6.2 The Charged Code Theorem: Statement and Examples
8.7 Bifix Codes in the Free Profinite Monoid
8.8 Proof of the Charged Code Theorem
8.8.1 A Byproduct of the Proof of the Charged Code Theorem
8.9 Exercises
8.9.1 Section 8.2
8.9.2 Section 8.3
8.9.3 Section 8.5
8.9.4 Section 8.6
8.9.5 Section 8.7
8.9.6 Section 8.8
8.10 Solutions
8.10.1 Section 8.2
8.10.2 Section 8.3
8.10.3 Section 8.5
8.10.4 Section 8.6
8.10.5 Section 8.7
8.10.6 Section 8.8
8.11 Notes
Bibliography
Subject Index
Index of symbols
