A Composition Law and Refined Notions of Convergence for Periodic Continued Fractions

We define an equivalence relation on periodic continued fractions with partial quotients in a ring O⊆C, a group law on these equivalence classes, and a map from these equivalence classes to matrices in GL2(O) with determinant ±1. We prove this group of equivalence classes is isomorphic to Z/2Z*O and study certain of its one- and two-dimensional representations. For a periodic continued fraction with period k, we give a refined description of the limits of the k different k-decimations of its sequence of convergents. We show that for a periodic continued fraction associated to a matrix with eigenvalues of different magnitudes, all k of these limits exist in P1(C) and a strict majority of them are equal.