On the Ordering and Injectivity of Lagrange Constants Associated with Certain Substitutions

Source abstract

Motivated by the modern characterization of the Markoff spectrum via mechanical words and the Markoff Uniqueness Conjecture, we study the Lagrange constants of continued fractions generated by a parameterized family of substitutions $φ_{a,b}$ $φ_{a,b}$ defined by $0 \mapsto aa$ and $1 \mapsto bb$ with $b = a + 1$. Based on a three-fold arithmetic classification of pairs of rational numbers $0 \le x a$, highlighting a sharp phase transition in ordering behavior at the boundary $b = a^2 + 2$.