BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Mélodie Lapointe (Université de Paris)
DTSTART:20211019T123000Z
DTEND:20211019T133000Z
DTSTAMP:20260423T052931Z
UID:OWNS/61
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OWNS/61/">q-
 analog of the Markoff injectivity conjecture</a>\nby Mélodie Lapointe (Un
 iversité de Paris) as part of One World Numeration seminar\n\n\nAbstract\
 nThe Markoff injectivity conjecture states that $w\\mapsto\\mu(w)_{12}$ is
  injective on the set of Christoffel words where $\\mu:\\{\\mathtt{0}\,\\m
 athtt{1}\\}^*\\to\\mathrm{SL}_2(\\mathbb{Z})$ is a certain homomorphism an
 d $M_{12}$ is the entry above the diagonal of a $2\\times2$ matrix $M$. Re
 cently\, Leclere and Morier-Genoud (2021) proposed a $q$-analog $\\mu_q$ o
 f $\\mu$ such that $\\mu_{q\\to1}(w)_{12}=\\mu(w)_{12}$ is the Markoff num
 ber associated to the Christoffel word $w$. We show that there exists an o
 rder $<_{radix}$ on $\\{\\mathtt{0}\,\\mathtt{1}\\}^*$ such that for every
  balanced sequence $s \\in \\{\\mathtt{0}\,\\mathtt{1}\\}^\\mathbb{Z}$ and
  for all factors $u\, v$ in the language of $s$ with $u <_{radix} v$\, the
  difference $\\mu_q(v)_{12} - \\mu_q(u)_{12}$ is a nonzero polynomial of i
 ndeterminate $q$ with nonnegative integer coefficients. Therefore\, for ev
 ery $q>0$\, the map $\\{\\mathtt{0}\,\\mathtt{1}\\}^*\\to\\mathbb{R}$ defi
 ned by $w\\mapsto\\mu_q(w)_{12}$ is increasing thus injective over the lan
 guage of a balanced sequence. The proof  uses an  equivalence between bala
 nced sequences satisfying some Markoff property and indistinguishable asym
 ptotic pairs.\n
LOCATION:https://researchseminars.org/talk/OWNS/61/
END:VEVENT
END:VCALENDAR