Generalized Continued Fraction Algorithm for the Index 3 Sublattice

Abstract

Motivated by an algorithm to generate all Pythagorean triples, Romik introduced a dynamical system on the unit circle, which corresponds the continued fraction algorithm on the index-2 sublattice. Cha et al. extended Romik’s work to other ellipses and spheres and developed a dynamical system generating all Eisenstein triples. In this article, we review the dynamical systems by Romik and by Cha et al. and find connections to the continued fraction algorithms.