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.
