The $q$-analog of the Markoff injectivity conjecture over the language of a balanced sequence

Abstract

The Markoff injectivity conjecture states that $w\mapsto \mu (w{)}_{12}$ is injective on the set of Christoffel words where $\mu :\{0,1{\}}^{\ast }\to {\mathrm{SL}}_2(\mathbb{Z})$ is a certain homomorphism and $M_{12}$ is the entry above the diagonal of a $2\times 2$ matrix $M$. Recently, Leclere and Morier-Genoud (2021) proposed a $q$-analog ${\mu }_q$ of $\mu$ such that ${\mu }_q(w{)}_{12}{|}_{q=1}=\mu (w{)}_{12}$ is the Markoff number associated to the Christoffel word $w$ when evaluated at $q=1$. We show that there exists an order ${<}_{radix}$ on $\{0,1{\}}^{\ast }$ such that for every balanced sequence $s\in \{0,1{\}}^{\mathbb{Z}}$ 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 indeterminate $q$ with nonnegative integer coefficients. Therefore, the map $u\mapsto {\mu }_q(u{)}_{12}$ is injective over the language of a balanced sequence. The proof uses an equivalence between balanced sequences satisfying some Markoff property and indistinguishable asymptotic pairs.

v1: 17 pages, 5 figures, 1 table; v2: 18 pages, restated results in terms of nonnegativity of coefficients of polynomials; v3: 20 pages, improvements made after reviews

References in corpus (1)

• A characterization of Sturmian sequences by indistinguishable asymptotic pairs