Combinatorics on words for Markoff numbers (Laurent Vuillon, Université de Savoie Mont Blanc)

01.04.2025 10:30

In a first part, we introduce the Markoff numberswhich 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 N^2 grid related to Markoff numbers. We show two conjectures of the famous book of Aigner by preserving themonotonicity of this valuation on a sequence of paths and byinvestigating the dynamic of local path transformations.

C. Lagisquet, E. Pelantová, S. Tavenas, L. Vuillon On the Markov numbers: fixed numerator, denominator, and sum conjectures Advances in Applied Mathematics 130, 102227 6 2021
C. Reutenauer and L. Vuillon, Palindromic Closures and Thue-Morse Substitution for Markoff Numbers, Uniform Distribution Theory 12 (2017), no. 2, 25-35.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-05, Tuesday 01.04.2025, Séminaire "Groupes et géométrie"

entrée libre