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.

Markoff’s Theorem for Approximations

This chapter provesMarkoff’s theorem for approximations: if x is an irrational real number such that its Lagrange number L(x) is <3, then the continued fraction of x is ultimately periodic and has as periodic pattern a Christoffel word written on the alphabet 11, 22. Moreover, the bound is attained: this means that there are indeed convergents whose error terms are correctly bounded. For this latter result, one needs a lot of technical results, which use the notion of good and bad approximation of a real number x satisfying L(x) <3: the ranks of the good and bad convergents are precisely given. These results are illustrated by the golden ratio and the number 1 + square root of 2.

Basics

Definitions and basic results about words: alphabet, length, free monoid, concatenation, prefix, suffix, factor, conjugation, reversal, palindrome, commutative image, periodicity, ultimate periodicity, periodic pattern, infinite words, bi-infinite words, free groups, reduced words, homomorphisms, embedding of a free monoid in a free group, abelianization,matrix of an endomorphism, GL2(Z), SL2(Z).

Lyndon Words and Christoffel Words

This chapter covers the lexicographical ordering of lower Christoffel words, which is equivalent to the ordering by their slopes (Borel and Laubie). Lower Christoffel words are particular Lyndon words. They are maximum for the lexicographical order among Lyndon words of a given slope (Borel and Laubie). They are, together with the upper Christoffel words, the only unbordered finite Sturmian words (Chuan). They are exactly the Lyndon words which are Sturmian words (Berstel and de Luca). The standard factorization of a lower Christoffel word is obtained by cutting before the smallest lexicographical suffix. Finally, they are exactly the Lyndon words which are equilibrated (Melançon).

Historical Notes

Short review of some earlier work on the classical Markoff theory, by Christoffel, Markoff, Hurwitz, Frobenius, Series, Smith, Bachmann, Remak, Dickson, Cassels, Cusick, Flahive, Heawood, Perron, Cassels, Bombieri, Aigner.

Lagrange Numbers Less Than Three

This chapter reflects the technical heart ofMarkoff’s theory. Themain theoremis that if the Lagrange number of a sequence is smaller than 3, then this sequence is the image under the substitution a->11, b->22 of a sequence satisfying theMarkoff property outlined inChapter 4. The proof goes through several technical lemmas: except for a finite number of terms, the sequence has only 1 and 2s; 121 and 212 are forbidden factors; the factors 1122 and 2211 are preceded and followed bywordswith a special lexicographical property; more forbidden factors; 1 and 2’s appear in pairs. The similar theorem for bi-infinite sequences is deduced.

The Markoff Property

The Markoff property is a combinatorial property of infinite words on the alphabet {a,b}, and of bi-infinite words. Such a word has this property if whenever there is a factor xy in the word,with x,y equal to the letters a,b (in some order), then itmay be extended into a factor of the formym’xymx, wherem’ is the reversal ofm, and where the length ofmis bounded (the bound depends only on the infinite word). As discussed in this chapter, the main theorem, due toMarkoff, is that this property implies periodicity, with a periodic pattern which must be a Christoffel word. It is one of the crucial results inMarkoff’s theory.

Words

Basic results on bases of the free abelian group Z2 are proved by elementary and geometric methods. Bases are characterized by determinants of matrices. Primitive triangles and primitive parallelograms are introduced and characterized by tiling of the plane. Christoffel words are introduced by discrete geometric means: they code paths which discretize finite segments with integral endpoints. Their inner maximal factor is a palindrome, which is the cutting word associated with the segment. Their standard factorization is studied; uniqueness is proved (Borel and Laubie). The tree of Christoffel pairs is studied using Sturmian morphisms: these are the substitutions that preserve conjugates of Christoffel words.

Bases and Automorphisms of the Free Group on Two Generators

The chapter begins with a self-contained exposition of the theory of Nielsen on the free groupwith two generators: bases of F(a, b),Nielsen’s criterion for automorphisms of F(a, b), It also coversNielsen’s theoremon abelianization of these automorphisms andWeinbaum’s theorem on representatives of the group of automorphisms modulo the subgroup of inner automorphism. Perrine’s theorem on bases of the derived group of SL2(Z) and Markoff triples is deduced, and a very simple and efficient algorithm for detecting bases of F(a, b) is given (Séébold, Kassel, the author). Positive automorphisms of F(a, b) are characterized (Wen andWen) and shown to coincide with Sturmian morphisms (Mignosi, Séébold).

Conjugates and Factors

Christoffel words are naturally cyclic objects. They may be defined by the Cayley graphs of finite cyclic groups. They have many characterizations through their conjugation classes; among them, one is obtained using the Burrows–Wheeler transform (Mantaci, Restivo, and Sciortino); another one is due to Pirillo. The sets of their circular factors have many remarkable properties; in particular the number of them of length k is k+1, if k is smaller than the length of the Christoffel word, and it is a characteristic property (Borel and the author), reminiscent of the similar property of Sturmian sequences. A related characterization, similar to that of Droubay and Pirillo for Sturmian sequences, rests on the count of palindromic factors. The set of finite Sturmian words, that is, the set of all factors of all Christoffel words, coincides with the set of balanced words (Dulucq and Gouyou–Beauchamps).