# From Christoffel Words to Markoff Numbers

@inproceedings{Reutenauer2018FromCW, title={From Christoffel Words to Markoff Numbers}, author={Christophe Reutenauer}, year={2018} }

Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, calledMarkoff numbers; they are characterized by a certain diophantine equality. More basically, they are constructed using certain words, essentially the…

## 21 Citations

Music and combinatorics on words: a historical survey

- ArtJournal of Mathematics and Music
- 2018

In the preface of Lothaire’s book Combinatorics On Words first published in 1983, Dominique Perrin wrote: “It can be roughly said that automata theory (and formal language theory) deals with sets of…

On the ordering of the Markov numbers

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The Markov numbers are the positive integers that appear in the solutions of the equation $x^2+y^2+z^2=3xyz$. These numbers are a classical subject in number theory and have important ramifications…

Equations and first-order theory of one-relator and word-hyperbolic monoids.

- Mathematics
- 2019

We investigate systems of equations and the first-order theory of one-relator monoids and of word-hyperbolic monoids. We describe a family of one-relator monoids of the form $\langle A\mid…

On the conjugates of Christoffel words

- Mathematics
- 2022

We introduce a parametrization of the conjugates of Christoffel words based on the integer Ostrowski numeration system. We use it to give a precise description of the borders (prefixes which are also…

Dimension Groups and Dynamical Systems

- MathematicsArXiv
- 2020

This book is the first self-contained exposition of the fascinating link between dynamical systems and dimension groups and the algebraic structures associated with them, with an emphasis on symbolic systems, particularly substitution systems.

The $q$-analog of the Markoff injectivity conjecture over the language of a balanced sequence

- Mathematics
- 2021

The Markoff injectivity conjecture states that w 7→ μ(w)12 is injective on the set of Christoffel words where μ : {0, 1}∗ → SL2(Z) is a certain homomorphism and M12 is the entry above the diagonal of…

All Growth Rates of Abelian Exponents Are Attained by Infinite Binary Words

- MathematicsMFCS
- 2020

It is obtained that every nonnegative real number is the critical abelian exponent of some infinite binary word.

q-Deformations in the modular group and of the real quadratic irrational numbers

- MathematicsAdv. Appl. Math.
- 2021