图书简介

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

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