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arXiv · 2504.02350

Inducing contractions of the mother of all continued fractions

Karma Dajani·Cor Kraaikamp·Slade Sanderson
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Abstract

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 $\alpha$-continued fractions for all $0<\alpha\le 1$ as examples of contracted Farey expansions. Moreover, we give a new description of a planar natural extension for each of the $\alpha$-continued fraction transformations as an explicit induced transformation of Ito's natural extension.

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Karma Dajani, Cor Kraaikamp, Slade Sanderson. 2025-04-03. Inducing contractions of the mother of all continued fractions. https://arxiv.org/abs/2504.02350

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