There is only one Farey map

Let $$A_0,A_1$$ be nonnegative matrices in $${{\,\textrm{GL}\,}}_{n+1}\mathbb {Z}$$ 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 $$n=1,2,\ldots $$ 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 $$G:\Delta \rightarrow \Delta $$ with branches $$A_0^{-1},A_1^{-1}$$ , which is continuous in exactly one of these three cases, namely when it equals the Farey-Mönkemeyer map.