A composition law and refined notions of convergence for periodic continued fractions

2023/07/06 by Bradley W. Brock, Brock, Bradley W., Bruce W. Jordan +3
Materials Science · Mathematics · #11J70 (Primary) 11G30 (Secondary) #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Liquid Crystal Research Advancements #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2307.02718

openalex publication_date 2023/07/06 · openalex created_date 2023/07/08 · openalex updated_date 2026/07/28

Abstract

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\astO 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.

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