Time reversed $S$-expansions
In 1991 a new class of continued fraction expansions, the $S$-expansions, was introduced in this journal. This class contains many classical continued fraction algorithms, such as Nakada’s $\alpha$-expansions (for $\alpha$ between $1/2$ and 1), the nearest integer continued fraction, Minkowski’s diagonal continued fraction expansion, and Bosma’s optimal continued fraction. These $S$-expansions were obtained from the natural extension of the regular continued fraction (RFC) via induced transformations. Therefore many metric and arithmetic properties of these $S$-expansions can be derived from the corresponding classical results on the RFC. In particular, the natural extensions of these $S$-expansions were obtained. The second coordinate map of these natural extensions is the inverse of a continued fraction algorithm. In this paper we study these ‘reversed algorithms’; in particular we show they are again $S$-expansions, and we find the corresponding singularization areas.