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.

For the first time in literature on the subject, this textbook treats the two aspects of Markoff's Theory simultaneously. Numerous figures throughout the book help to illustrate its points and provide proofs of discrete geometry.

• 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.

For the first time in literature on the subject, this textbook treats the two aspects of Markoff's Theory simultaneously. Numerous figures throughout the book help to illustrate its points and provide proofs of discrete geometry.

• 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