Aix-Marseille Université
Institut de Mathématiques de Marseille (I2M) - UMR 7373
3 place Victor Hugo
Case 19
13331 Marseille Cedex 3


Combinatorics on words for Markoff numbers (Laurent Vuillon)

Laurent Vuillon

Date(s) : 11/06/2024   iCal
11h00 - 12h00

In a first part, we introduce the Markoff numbers, which are fascinating integers related to number theory, Diophantine equation, hyperbolic geometry, continued fractions and Christoffel words. Many great mathematicians have worked on these numbers, and the 100 years uniqueness conjecture by Frobenius is still unsolved. We state a new formula to compute the Markoff numbers, using iterated palindromic closure and the Thue-Morse substitution. The main theorem shows that for each Markoff number m, there exists a word v∈{a,b}∗ such that m−2 is equal to the length of the iterated palindromic closure of the iterated antipalindromic closure of the word av. This work gives a recursive construction of Markoff numbers by the lengths of words generated by a discrete dynamical system.

In a second part, we show new results on Markoff numbers by using generalizations of Farey’s fractions and Christoffel words. In particular, we study a valuation of paths on the ℕ² grid related to Markoff numbers. We show two conjectures of the famous book by Aigner, by preserving the monotonicity of this valuation on a sequence of paths, and by investigating the dynamic of local path transformations.

C. Reutenauer and L. Vuillon, Palindromic Closures and Thue-Morse Substitution for Markoff Numbers, Uniform Distribution Theory 12(2), (2017),25–35.

C. Lagisquet, E. Pelantová, S. Tavenas and L. Vuillon « On the Markov numbers : fixed numerator, denominator, and sum conjectures ». Advances in Applied Mathematics, 130, 102227, (2021).

Site Sud, Luminy, TPR2, Salle de Séminaire 304-306 (3ème étage)


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