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.
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