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The \(q\)-analog of the Markoff injectivity conjecture over the language of a balanced sequence

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The Markoff injectivity conjecture states that \(w\mapsto\mu(w)_{12}\) is injective on the set of Christoffel words where \(\mu:\{\mathtt{0},\mathtt{1}\}^*\to\mathrm{SL}_2(\mathbb{Z})\) is a certain homomorphism and \(M_{12}\) is the entry above the diagonal of a \(2\times2\) 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 \(\{\mathtt{0},\mathtt{1}\}^*\) 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 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.

Mathematics Subject Classifications: 11J06, 68R15, 05A30

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