Paper056_The_Calta_Matrix

ABSTRACT. We define the Calta Matrix Cm(x, α) = ∏j=0m−1 AkjClj ∈ SL(2, 𝒪Kn) for the α-continued fraction on Gn = Δ(3, n, ∞). We derive every dynamical property in [1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11] from det(Cm) = 1: bijectivity from invertibility, the invariant measure from the chain rule, matching from the quadratic formula over 𝒪Kn, entropy from the characteristic polynomial, and the expansive power from an iteration bound on the bottom row. We then resolve all conjectures (Conjectures 6, 8, 147), all open questions (Questions 1–3), and the implicit conjecture on extension to Gm,n = Δ(m, n, ∞) posed in [1].