In 1875, Elwin Bruno Christoffel introduced a special class of words on a binary alphabet linked to continued fractions which would go onto be known as Christoffel words. Some years later, Andrey Markoff published his famous theory, the now called Markoff theory. It characterized certain quadratic forms and certain real numbers by extremal inequalities. Both classes are constructed using certain natural numbers known as Markoff numbers and they are characterized by a certain Diophantine equality. More basically, they are constructed using certain words essentially the Christoffel words. The link between Christoffel words and the theory of Markoff was noted by Ferdinand Frobenius in 1913, but has been neglected in recent times. Motivated by this overlooked connection, this book looks to expand on the relationship between these two areas. Part 1 focuses on the classical theory of Markoff, while Part II explores the more advanced and recent results of the theory of Christoffel words.
Author(s): Christophe Reutenauer
Publisher: Oxford University Press
Year: 2019
Language: English
Commentary: To avoid font issues, please use https://www.sejda.com/
Pages: 169
Cover
From Christoffel Words to Markoff Numbers
Copyright
Dedication
Acknowledgements
Contents
Introduction
PART I The Theory of Markoff
1
Basics
2 Words
2.1 Tiling the Plane with a Parallelogram
2.2 Christoffel Words
2.3 Palindromes
2.4 Standard Factorization
2.5 The Tree of Christoffel Pairs
2.6 Sturmian Morphisms
3 Markoff Numbers
3.1 Markoff Triples and Numbers
3.2 The Tree of Markoff Triples
3.3 The Markoff Injectivity Conjecture
4 The Markoff Property
4.1 Markoff Property for Infinite Words
4.2 Markoff Property for Bi-infinite Words
5 Continued Fractions
5.1 Finite Continued Fractions
5.2 Infinite Continued Fractions
5.3 Periodic Expansions Yield Quadratic Numbers
5.4 Approximations of Real Numbers
5.5 Lagrange Number of a Real Number
5.6 Ordering Continued Fractions
6 Words and Quadratic Numbers
6.1 Continued Fractions Associated with Christoffel Words
6.2 Markoff Supremum of a Bi-infinite Sequence
6.3 Lagrange Number of a Sequence
7 Lagrange Numbers Less Than Three
7.1 FromL(s) < 3 to the Markoff Property
7.2 Bi-infinite Sequences
8 Markoff’s Theorem for Approximations
8.1 Main Lemma
8.2 Markoff’s Theorem for Approximations
8.3 Good and Bad Approximations
9 Markoff’s Theorem for Quadratic Forms
9.1 Indefinite Real Binary Quadratic Forms
9.2 Infimum
9.3 Markoff’s Theorem for Quadratic Forms
10 Numerology
10.1 Thirteen Markoff Numbers
10.2 The Golden Ratio and Other Numbers
10.3 The Matrices μ(w) and Frobenius Congruences
10.4 Markoff Quadratic Forms
11 Historical Notes
PART II The Theory of Christoffel Words
12 Palindromes and Periods
12.1 Palindromes
12.2 Periods
13 Lyndon Words and Christoffel Words
13.1 Slopes
13.2 Lyndon Words
13.3 Maximal Lyndon Words
13.4 Unbordered Sturmian Words
13.5 Equilibrated Lyndon Words
14 Stern–Brocot Tree
14.1 The Tree of Christoffel Words
14.2 Stern–Brocot Tree and Continued Fractions
14.3 The Raney Tree and Dual Words
14.4 Convex Hull
15 Conjugates and Factors
15.1 Cayley Graph
15.2 Conjugates
15.3 Factors
15.4 Palindromes Again
15.5 Finite Sturmian Words
16 Bases and Automorphisms of the Free Group on Two Generators
16.1 Bases and Automorphisms of F(a,b)
16.2 Inner Automorphisms
16.3 Christoffel Bases of F(a,b)
16.4 Nielsen’s Criterion
16.5 An Algorithm for the Bases of F(a,b)
16.6 Sturmian Morphisms Again
17 Complements
17.1 Other Results on Christoffel Words
17.2 Lyndon Words and Lie Theory
17.3 Music
Bibliography
Index
