Purely periodic continued fractions and graph-directed iterated function systems

TL;DR

This paper introduces a geometric framework linking Gauss-type maps with graph-directed iterated function systems, enabling explicit characterization of purely periodic continued fractions for various algorithms.

Contribution

It develops a unified approach to represent multiple continued fraction algorithms through graph-directed systems and characterizes quadratic irrationals with purely periodic expansions.

Findings

Unified geometric realization of Gauss-type maps

Explicit dual algorithms for continued fractions

Characterization of purely periodic quadratic irrationals

Abstract

We describe Gauss-type maps as geometric realizations of certain codes in the monoid of nonnegative matrices in the extended modular group. Each such code, together with an appropriate choice of unimodular intervals in P^1R, determines a dual pair of graph-directed iterated function systems, whose attractors contain intervals and constitute the domains of a dual pair of Gauss-type maps. Our framework covers many continued fraction algorithms (such as Farey fractions, Ceiling, Even and Odd, Nearest Integer, ...) and provides explicit dual algorithms and characterizations of those quadratic irrationals having a purely periodic expansion.