There is only one Farey map
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908391645904896 |
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| author | Panti, Giovanni |
| author_facet | Panti, Giovanni |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_02984 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | There is only one Farey map Panti, Giovanni Dynamical Systems Number Theory 11J70, 37M25 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. |
| title | There is only one Farey map |
| topic | Dynamical Systems Number Theory 11J70, 37M25 |
| url | https://arxiv.org/abs/2506.02984 |