Geodesic flows and the mother of all continued fractions
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arXiv
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| Format: | Preprint |
| Publié: |
2020
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| _version_ | 1866914845363798016 |
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| author | Merriman, Claire |
| author_facet | Merriman, Claire |
| contents | We extend the Series' connection between the modular surface $\mathcal{M}=\operatorname{PSL}(2,\mathbb{Z})\backslash\mathbb{H}$, cutting sequences, and regular continued fractions to the slow converging Lehner and Farey continued fractions with digits $(1,+1)$ and $(2,-1)$ in the notation used for the Lehner continued fractions. We also introduce an alternative insertion and singularization algorithm for Farey expansions and other non-semiregular continued fractions, and an alternative dual expansion to the Farey expansions so that $\frac{dxdy}{(1+xy)^2}$ is invariant under the natural extension map. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2001_06073 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Geodesic flows and the mother of all continued fractions Merriman, Claire Dynamical Systems Number Theory We extend the Series' connection between the modular surface $\mathcal{M}=\operatorname{PSL}(2,\mathbb{Z})\backslash\mathbb{H}$, cutting sequences, and regular continued fractions to the slow converging Lehner and Farey continued fractions with digits $(1,+1)$ and $(2,-1)$ in the notation used for the Lehner continued fractions. We also introduce an alternative insertion and singularization algorithm for Farey expansions and other non-semiregular continued fractions, and an alternative dual expansion to the Farey expansions so that $\frac{dxdy}{(1+xy)^2}$ is invariant under the natural extension map. |
| title | Geodesic flows and the mother of all continued fractions |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2001.06073 |