Numerology

This chapter gives several examples, which may help the reader to work in concrete terms with Markoff numbers, Christoffel words, Markoff constants, and quadratic forms. In particular the thirteen Markoff numbers <1000 are given, together with the associated mathematical objects considered before in the book:Markoff constants, Christoffel words, the associated matrices by the representation of Chapter 3, theMarkoff quadratic numbers whose expansion is given by the Christoffel word, the Markoff quadratic forms. Some results of Frobenius, Aigner, andClemens are given. In particular thematrix associated with a Christoffel word may be computed directly from its Markoff triple.

From Christoffel Words to Markoff Numbers

Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, calledMarkoff numbers; they are characterized by a certain diophantine equality. More basically, they are constructed using certain words, essentially the Christoffel words. The link between Christoffelwords and the theory ofMarkoffwas noted by Frobenius.Motivated by this link, the book presents the classical theory of Markoff in its two aspects, based on the theory of Christoffel words. This is done in Part I of the book. Part II gives the more advanced and recent results of the theory of Christoffel words: palindromes (central words), periods, Lyndon 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.

Markoff Numbers

The Markoff equation is the diophantine equation x2 +y2 +z2 = 3xyz. A solution is called a Markoff triple. The main result in this chapter is a bijection between lower Christoffel words and Markoff triples. The bijection uses several ingredients: a special representation of the free monoid into SL2(N), the so-called Fricke relations, which relate the traces of two matrices in SL2, their product and their commutator (an equation reminiscent of the Markoff equation, as noted first by Harvey Cohn). Another lemma describes the socalled Markoff moves: they relate Markoff triples each to another. The chapter ends with a statement of the famous Frobenius conjecture: it asks whether the parametrization of Markoff numbers (that is, components of a Markoff triple), which is surjective by the theorem, is also injective.

Palindromic Closures and Thue-Morse Substitution for Markoff Numbers

We state a new formula to compute the Markoff numbers using iterated palindromic closure and the Thue-Morse substitution. The main theorem shows that for each Markoff number m, there exists a word v ∈ {a, b}∗ such that m − 2 is equal to the length of the iterated palindromic closure of the iterated antipalindromic closure of the word av. This construction gives a new recursive construction of the Markoff numbers by the lengths of the words involved in the palindromic closure. This construction interpolates between the Fibonacci numbers and the Pell numbers.