Profinite Semigroups and Symbolic Dynamics

1. Front Matter

   Pages i-ix

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2. Introduction

   • Jorge Almeida, Alfredo Costa, Revekka Kyriakoglou, Dominique Perrin

   Pages 1-5

3. Prelude: Profinite Integers

   • Jorge Almeida, Alfredo Costa, Revekka Kyriakoglou, Dominique Perrin

   Pages 7-23

4. Profinite Groups and Semigroups

   • Jorge Almeida, Alfredo Costa, Revekka Kyriakoglou, Dominique Perrin

   Pages 25-85

5. Free Profinite Monoids, Semigroups and Groups

   • Jorge Almeida, Alfredo Costa, Revekka Kyriakoglou, Dominique Perrin

   Pages 87-137

6. Shift Spaces

   • Jorge Almeida, Alfredo Costa, Revekka Kyriakoglou, Dominique Perrin

   Pages 139-168

7. Sturmian Sets and Tree Sets

   • Jorge Almeida, Alfredo Costa, Revekka Kyriakoglou, Dominique Perrin

   Pages 169-194

8. The Schützenberger Group of a Minimal Set

   • Jorge Almeida, Alfredo Costa, Revekka Kyriakoglou, Dominique Perrin

   Pages 195-214

9. Groups of Bifix Codes

   • Jorge Almeida, Alfredo Costa, Revekka Kyriakoglou, Dominique Perrin

   Pages 215-263

10. Back Matter

    Pages 265-278

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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.

• Minimal Cantor Systems
• Profinite Integers
• Profinite Semigroups and Groups
• Substitution Dynamical Systems
• Symbolic Dynamics

• CMUP, Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, Porto, Portugal

  Jorge Almeida

• CMUC, Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade de Coimbra, Coimbra, Portugal

  Alfredo Costa

• LIGM, Universite Gustave Eiffel, Marne La Vallee Cedex 2, France

  Revekka Kyriakoglou, Dominique Perrin

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