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Time reversed $S$-expansions

Tom 190 / 2019

Cor Kraaikamp, Irene Ravesloot Acta Arithmetica 190 (2019), 363-380 MSC: Primary 11J70; Secondary 11K50. DOI: 10.4064/aa180525-4-2 Opublikowany online: 5 August 2019

Streszczenie

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.

Autorzy

  • Cor KraaikampTechnische Universiteit Delft
    EWI, DIAM
    Van Mourik Broekmanweg 6
    2628 XE Delft, The Netherlands
    e-mail
  • Irene RaveslootTechnische Universiteit Delft
    EWI, DIAM
    Van Mourik Broekmanweg 6
    2628 XE Delft, The Netherlands
    e-mail

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