From Christoffel Words to Markoff Numbers
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Product details:
- Edition number 2
- Publisher OUP Oxford
- Date of Publication 23 July 2026
- ISBN 9780197907559
- Binding Hardback
- No. of pages272 pages
- Size 234x156 mm
- Language English
- Illustrations 27 black & white and colour figures 700
Categories
Short description:
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.
MoreLong description:
In 1875, Elwin Bruno Christoffel introduced a special class of words on a binary alphabet linked to continued fractions which would go on to be known as Christoffel words. Some years later, Andrey Markoff published his famous theory, now called the Markoff theory. Markoff's theory characterized certain quadratic forms and real numbers by extremal inequalities. Both classes are constructed using certain natural numbers, known as Markoff numbers, and they form part of a solution to the Markoff Diophantine equation. More basically, they are constructed using certain words, essentially Christoffel words.
The link between Christoffel words and Markoff's theory 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. The first part of the book focuses on the classical theory of Markoff, while Part II explores the more advanced and recent results of the theory of Christoffel words.
This new edition includes many additional exercises and solutions, as well as expanded sections on quadratic forms and quadratic numbers, two new chapters on standard words and the commutator subgroup, and revised results in combinatorics on words.
Review from previous edition 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.
Table of Contents:
Part I The theory of Markoff
Basics
Words
Markoff numbers
The Markoff property
Continued fractions
Binary quadratic forms
Lagrange number
Words and quadratic numbers
Lagrange numbers less than three
Markoff's theorem for approximations
Markoff's theorem for quadratic forms
Numerology
Historical notes
Part II The theory of Christoffel words
Palindromes and periods
Conjugates and factors
Standard words
Lyndon words and Christoffel words
Sturmian words
Stern-Brocot tree
Free group on two generators
The commutator subgroup SL2(Z)'
