Commensurable continued fractions

TL;DR

This paper compares two algebraic continued fraction algorithms, showing their natural extensions are conjugate to geodesic flow return maps and that most real numbers share infinitely many approximants for both.

Contribution

It provides explicit models of natural extensions, proves their conjugacy to geodesic flow return maps, and establishes common approximants for almost all real numbers.

Findings

Natural extensions are conjugate to geodesic flow return maps

Almost every real number has infinitely many common approximants

Explicit models of the natural extensions are constructed

Abstract

We compare two families of continued fractions algorithms, the symmetrized Rosen algorithm and the Veech algorithm. Each of these algorithms expands real numbers in terms of certain algebraic integers. We give explicit models of the natural extension of the maps associated with these algorithms; prove that these natural extensions are in fact conjugate to the first return map of the geodesic flow on a related surface; and, deduce that, up to a conjugacy, almost every real number has an infinite number of common approximants for both algorithms.