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UID:5180@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20240611T110000
DTEND;TZID=Europe/Paris:20240611T120000
DTSTAMP:20240819T140154Z
URL:https://www.i2m.univ-amu.fr/evenements/combinatorics-on-words-for-mark
 off-numbers/
SUMMARY:Laurent Vuillon (Chambéry): Combinatorics on words for Markoff num
 bers
DESCRIPTION:Laurent Vuillon: In a first part\, we introduce the Markoff num
 bers\, which are fascinating integers related to number theory\, Diophanti
 ne equation\, hyperbolic geometry\, continued fractions and Christoffel wo
 rds. 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 closu
 re and the Thue-Morse substitution. The main theorem shows that for each M
 arkoff number m\, there exists a word v∈{a\,b}∗ such that m−2 is equ
 al to the length of the iterated palindromic closure of the iterated antip
 alindromic closure of the word av. This work gives a recursive constructio
 n of Markoff numbers by the lengths of words generated by a discrete dynam
 ical system.\n\nIn 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 Ma
 rkoff numbers. We show two conjectures of the famous book by Aigner\, by p
 reserving the monotonicity of this valuation on a sequence of paths\, and 
 by investigating the dynamic of local path transformations.\n\nC. Reutenau
 er and L. Vuillon\, Palindromic Closures and Thue-Morse Substitution for M
 arkoff Numbers\, Uniform Distribution Theory 12(2)\, (2017)\,25–35.\n\nC
 . Lagisquet\, E. Pelantová\, S. Tavenas and L. Vuillon "On the Markov num
 bers : fixed numerator\, denominator\, and sum conjectures". Advances in A
 pplied Mathematics\, 130\, 102227\, (2021).
CATEGORIES:Séminaire,Ernest
LOCATION:I2M Luminy - TPR2\, Salle de Séminaire 304-306 (3ème étage)\, 1
 63 Avenue de Luminy\, Marseille\, 13009\, France
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 e de Séminaire 304-306 (3ème étage):geo:0,0
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