Inducing contractions of the mother of all continued fractions

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 ([19]) 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 $0<\alpha \le 1$ 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.