There is only one Farey map

Let be nonnegative matrices in such that the subsimplexes split the standard unit n-dimensional simplex in two. We prove that, for every and up to the natural action of the symmetric group by conjugation, there are precisely three choices for the pair 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 with branches, which is continuous in exactly one of these three cases, namely when it equals the Farey-Mönkemeyer map.