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E-grāmata: From Christoffel Words to Markoff Numbers

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(Professor of Mathematics, Université du Québec ą Montréal)
  • Formāts: 208 pages
  • Izdošanas datums: 01-Nov-2018
  • Izdevniecība: Oxford University Press
  • Valoda: eng
  • ISBN-13: 9780192562593
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From Christoffel Words to Markoff NumbersPalielināt
  • Formāts: 208 pages
  • Izdošanas datums: 01-Nov-2018
  • Izdevniecība: Oxford University Press
  • Valoda: eng
  • ISBN-13: 9780192562593
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In 1875, Elwin Bruno Christoffel introduced a special class of words on a binary alphabet linked to continued fractions which would go onto be known as Christoffel words. Some years later, Andrey Markoff published his famous theory, the now called Markoff theory. It characterized certain quadratic forms and certain real numbers by extremal inequalities. Both classes are constructed using certain natural numbers known as Markoff numbers and 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 Ferdinand Frobenius in 1913, but has been neglected in recent times. Motivated by this overlooked connection, this book looks to expand on the relationship between these two areas. Part 1 focuses on the classical theory of Markoff, while Part II explores the more advanced and recent results of the theory of Christoffel words.

Recenzijas

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. * Ioan Tomescu, zbMath * This book is a well-written introduction to a fascinating subtopic in mathematics that should be accessible to a graduate student or a research mathematician. Most subsections are a few pages in length, easily digestible in a brief time. The proofs are concise, but with enough detail to satisfy the reader. * Jonathan Bayless, Associate Professor and Chair of Mathematics, Husson University, MAA Reviews *

Introduction 1(6)
PART I THE THEORY OF MARKOFF
1 Basics
7(2)
2 Words
9(12)
2.1 Tiling the Plane with a Parallelogram
9(2)
2.2 Christoffel Words
11(1)
2.3 Palindromes
12(2)
2.4 Standard Factorization
14(3)
2.5 The Tree of Christoffel Pairs
17(1)
2.6 Sturmian Morphisms
18(3)
3 Markoff Numbers
21(8)
3.1 Markoff Triples and Numbers
21(4)
3.2 The Tree of Markoff Triples
25(1)
3.3 The Markoff Injectivity Conjecture
26(3)
4 The Markoff Property
29(6)
4.1 Markoff Property for Infinite Words
29(3)
4.2 Markoff Property for Bi-infinite Words
32(3)
5 Continued Fractions
35(8)
5.1 Finite Continued Fractions
35(1)
5.2 Infinite Continued Fractions
36(1)
5.3 Periodic Expansions Yield Quadratic Numbers
37(1)
5.4 Approximations of Real Numbers
38(2)
5.5 Lagrange Number of a Real Number
40(1)
5.6 Ordering Continued Fractions
41(2)
6 Words and Quadratic Numbers
43(6)
6.1 Continued Fractions Associated with Christoffel Words
43(1)
6.2 Markoff Supremum of a Bi-infinite Sequence
44(3)
6.3 Lagrange Number of a Sequence
47(2)
7 Lagrange Numbers Less Than Three
49(6)
7.1 From L(s) <3 to the Markoff Property
49(3)
7.2 Bi-infinite Sequences
52(3)
8 Markoff's Theorem for Approximations
55(8)
8.1 Main Lemma
55(1)
8.2 Markorff's Theorem for Approximations
55(3)
8.3 Good and Bad Approximations
58(5)
9 Markoffs Theorem for Quadratic Forms
63(6)
9.1 Indefinite Real Binary Quadratic Forms
63(2)
9.2 Infimum
65(2)
9.3 Markoffs Theorem for Quadratic Forms
67(2)
10 Numerology
69(6)
10.1 Thirteen Markoff Numbers
69(1)
10.2 The Golden Ratio and Other Numbers
70(1)
10.3 The Matrices μ(w) and Frobenius Congruences
71(2)
10.4 Markoff Quadratic Forms
73(2)
11 Historical Notes
75(4)
PART III THE THEORY OF CHRISTOFFEL WORDS
12 Palindromes and Periods
79(8)
12.1 Palindromes
79(4)
12.2 Periods
83(4)
13 Lyndon Words and Christoffel Words
87(8)
13.1 Slopes
87(1)
13.2 Lyndon Words
88(1)
13.3 Maximal Lyndon Words
89(1)
13.4 Unbordered Sturmian Words
90(2)
13.5 Equilibrated Lyndon Words
92(3)
14 Stern-Brocot Tree
95(14)
14.1 The Tree of Christoffel Words
95(3)
14.2 Stern-Brocot Tree and Continued Fractions
98(3)
14.3 The Raney Tree and Dual Words
101(3)
14.4 Convex Hull
104(5)
15 Conjugates and Factors
109(16)
15.1 Cayley Graph
109(2)
15.2 Conjugates
111(4)
15.3 Factors
115(3)
15.4 Palindromes Again
118(3)
15.5 Finite Sturmian Words
121(4)
16 Bases and Automorphisms of the Free Group on Two Generators
125(18)
16.1 Bases and Automorphisms of F(a, b)
125(4)
16.2 Inner Automorphisms
129(3)
16.3 Christoffel Bases of F(a, b)
132(3)
16.4 Nielsen's Criterion
135(2)
16.5 An Algorithm for the Bases of F(a,b)
137(2)
16.6 Sturmian Morphisms Again
139(4)
17 Complements
143(6)
17.1 Other Results on Christoffel Words
143(3)
17.2 Lyndon Words and Lie Theory
146(1)
17.3 Music
146(3)
Bibliography 149(6)
Index 155
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 ą 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.
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