From Christoffel Words to Markoff Numbers

ISBN : 9780198827542

Christophe Reutenauer
208 Pages
156 x 234 mm
Pub date
Nov 2018
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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.


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 to Christoffel words
6.2 Marko supremum of a bi-innite sequence
6.3 Lagrange number of a sequence
7 Lagrange numbers less than 3
7.1 From L(s) 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 U(w) and Frobenius congruences
10.4 Markoff quadratic forms
11 Historical notes
The Theory of Christoel 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 Free group on two generators
16.1 Bases and automorphisms
16.2 Inner automorphisms
16.3 Christoffel bases
16.4 Nielsen's criterion
16.5 An algorithm for the bases
16.6 Sturmian morphisms again
17 Complements
17.1 Other results on Christoffel words
17.2 Lyndon words and Lie theory
17.3 Music

About the author: 

Christophe Reutenauer was educated at the Universite Paris in 1977 before going on to complete his doctorate thesis at the same institution in 1980. He was a former researcher at CNRS (Centre National de la Recherche Scientifique) in Paris and LITP (Laboratoire d'Informatique Theorique et de Programmation) from 1976 to 1990. Reutenauer has, from 1985, been a professor at UQAM (Universite du Quebec a Montreal), and was also a professor at the University of Strasbourg between 1999 and 2001. Since then, he has been an invited professor or researcher at several universities, including Saarbrucken, Darmstadt, Roma, Napoli, Palermo, UQAM, San Diego (UCSD), Strasbourg, Montpelier, Bordeaux, Paris-Est, Nice, and the Mittag-Leffler Institute. He was also the Canadian Research Chair for Algebra, Combinatorics and mathematical Informatics between 2001 and 2015.

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