Inducing contractions of the mother of all continued fractions

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 $α$-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.