There is only one Farey map

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.