This book looks to expand on the relationship between Christoffel words and Markoff theory. Part 1 focuses on the classical theory of Markoff, while part II explores the more advanced and recent results around 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) < 3 to the Marko 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 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

Christophe Reutenauer was educated at the Université 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 Théorique et de Programmation) from 1976 to 1990.
Reutenauer has, from 1985, been a professor at UQAM (Université du Québec à Montréal), 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 Saarbrücken, 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.