There is only one Farey map

TL;DR

This paper classifies all pairs of matrices that generate a topologically contractive projective iterated function system on an n-simplex, revealing exactly three such algorithms in each dimension, including the Farey-Monkemeyer map.

Contribution

It proves that, up to symmetry, only three pairs of matrices produce a topologically contractive system, leading to three unique continued fraction algorithms in each dimension.

Findings

Exactly three such matrix pairs exist in each dimension.

Among these, only the Farey-Monkemeyer map is continuous.

These algorithms produce distinct two-symbol expansions for points.

Abstract

Let A_0, A_1 be nonnegative matrices in GL(n+1,Z) such that the subsimplexes A_0[Delta], A_1[Delta] split the standard unit n-dimensional simplex Delta in two. We prove that, for every n=1,2,... and up to the natural action of the symmetric group by conjugation, there are precisely three choices for the pair (A_0, A_1) such that the resulting projective Iterated Function System is topologically contractive. In equivalent terms, in every dimension there exist precisely three continued fraction algorithms that assign distinct two-symbol expansions to distinct points. These expansions are induced by the Gauss-type map G: Delta --> Delta with branches A_0^{-1}, A_1^{-1}, which is continuous in exactly one of these three cases, namely when it equals the Farey-Monkemeyer map.