From Christoffel Words to Markoff Numbers

Abstract

Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years later Markoff published his famous theory, called now Markoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, called Markoff numbers; 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 Frobenius. Motivated by this link, the book presents classical theory of Markoff in its two aspects, based on the theory of Christoffel words. This is done in Part I of the book. In Part II is given the more advanced and recent results of the theory of Christoffel words: palindromes (central words), periods, Lyndon words, standard words, finite Sturmian words, Stern-Brocot tree, semi-convergents of rational numbers and finite continued fractions, geometric interpretations, conjugation, factors of Christoffel words, finite Sturmian words, free group on two generators, bases, inner automorphisms, Christoffel bases, Nielsen’s criterion, Sturmian morphisms and positive automorphisms of this free group, commutator subgroup, its bases and their relation to Markoff triples.