Inducing contractions of the mother of all continued fractions
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915225338380288 |
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| author | Dajani, Karma Kraaikamp, Cor Sanderson, Slade |
| author_facet | Dajani, Karma Kraaikamp, Cor Sanderson, Slade |
| contents | We introduce a new, large class of continued fraction algorithms producing what are called contracted Farey expansions. These algorithms are defined by coupling two acceleration techniques -- induced transformations and contraction -- in the setting of Shunji Ito's natural extension of the Farey tent map, which generates `slow' continued fraction expansions. In addition to defining new algorithms, we also realise several existing continued fraction algorithms in our unifying setting. In particular, we find regular continued fractions, the second-named author's $S$-expansions, and Nakada's parameterised family of $α$-continued fractions for all $0<α\le 1$ as examples of contracted Farey expansions. Moreover, we give a new description of a planar natural extension for each of the $α$-continued fraction transformations as an explicit induced transformation of Ito's natural extension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02350 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inducing contractions of the mother of all continued fractions Dajani, Karma Kraaikamp, Cor Sanderson, Slade Number Theory Dynamical Systems 11A55 (Primary) 37A05, 37A44 (Secondary) We introduce a new, large class of continued fraction algorithms producing what are called contracted Farey expansions. These algorithms are defined by coupling two acceleration techniques -- induced transformations and contraction -- in the setting of Shunji Ito's natural extension of the Farey tent map, which generates `slow' continued fraction expansions. In addition to defining new algorithms, we also realise several existing continued fraction algorithms in our unifying setting. In particular, we find regular continued fractions, the second-named author's $S$-expansions, and Nakada's parameterised family of $α$-continued fractions for all $0<α\le 1$ as examples of contracted Farey expansions. Moreover, we give a new description of a planar natural extension for each of the $α$-continued fraction transformations as an explicit induced transformation of Ito's natural extension. |
| title | Inducing contractions of the mother of all continued fractions |
| topic | Number Theory Dynamical Systems 11A55 (Primary) 37A05, 37A44 (Secondary) |
| url | https://arxiv.org/abs/2504.02350 |