On a tree of rational functions related to continued fractions
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arXiv
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| Format: | Preprint |
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2025
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| author | Langeveld, Niels Ralston, David |
| author_facet | Langeveld, Niels Ralston, David |
| contents | In this article, we present a binary tree with vertices given by rational functions $p(x)/q(x)$; the root and functional derivation of children are inspired by continued fractions. We prove some special properties of the tree. For example, the zero solutions of the denominators $q(x)$ are all real negative numbers and are dense in $(-\infty,-1]$. For $x>0$ functions are non intersecting and form a dense subset of $(0,1)$. Furthermore, when evaluating the tree for positive rational values, the tree contains every rational in $(0,1)$ exactly once if and only if $x\in \mathbb{N}$. For $x=1$, one finds back the classical Farey tree which is related to regular continued fractions. In the last part, we will make a similar tree in a similar way but for backward continued fractions. We highlight some similarities and differences. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_11595 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a tree of rational functions related to continued fractions Langeveld, Niels Ralston, David Dynamical Systems Number Theory 11J70, 37E05, 37A44 In this article, we present a binary tree with vertices given by rational functions $p(x)/q(x)$; the root and functional derivation of children are inspired by continued fractions. We prove some special properties of the tree. For example, the zero solutions of the denominators $q(x)$ are all real negative numbers and are dense in $(-\infty,-1]$. For $x>0$ functions are non intersecting and form a dense subset of $(0,1)$. Furthermore, when evaluating the tree for positive rational values, the tree contains every rational in $(0,1)$ exactly once if and only if $x\in \mathbb{N}$. For $x=1$, one finds back the classical Farey tree which is related to regular continued fractions. In the last part, we will make a similar tree in a similar way but for backward continued fractions. We highlight some similarities and differences. |
| title | On a tree of rational functions related to continued fractions |
| topic | Dynamical Systems Number Theory 11J70, 37E05, 37A44 |
| url | https://arxiv.org/abs/2512.11595 |