Abstract
This is a survey on the theory of Markoff, in its two aspects: quadratic forms (the original point of view of Markoff), approximation of reals. A link wih combinatorics on words is shown, through the notion of Christoffel words and special palindromes, called central words. Markoff triples may be characterized, by using some linear representation of the free monoid, restricted to these words, and Fricke relations. A double iterated palindromization allows to construct all Markoff numbers and to reformulate the Markoff numbers injectivity conjecture (Frobenius, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin 26:458–487, 1913).
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Notes
- 1.
The free monoid A ∗ is the set of words (= strings = finite sequences) on the set A, including the empty one; this is a monoid, the product of two words being the concatenation.
- 2.
This means that w = uv and \(\tilde w=vu\) for some words u, v.
- 3.
Frobenius states it, in two different forms, as an open problem, not a conjecture [19] p. 601 and 614.
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Reutenauer, C. (2018). Combinatorics on Words and the Theory of Markoff. In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K., Munthe-Kaas, H. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-01593-0_24
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