From Christoffel Words to Markoff Numbers [Hardcover]

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.

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 lã!