Natural extensions for Nakada's α-expansions

Descending from 1 to g<sup>2</sup>

Journal article (2018)

Authors

C.J. de Jonge Applied Probability - EEMCS , Universiteit van Amsterdam
C. Kraaikamp Applied Probability - EEMCS
Research Group
Applied Probability (EEMCS) (TU Delft)
Copyright: © 2018 C.J. de Jonge, C. Kraaikamp
DOI: https://doi.org/10.1016/j.jnt.2017.07.012
Continued fractions Metric theory
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Published Date

2018-2

Language

English

Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Applied Probability

Abstract

By means of singularisations and insertions in Nakada's α-expansions, which involves the removal of partial quotients 1 while introducing partial quotients with a minus sign, the natural extension of Nakada's continued fraction map Tα is given for (10-2)/3≤α<1. From our construction it follows that Ωα, the domain of the natural extension of Tα, is metrically isomorphic to Ωg for α∈[g2,g), where g is the small golden mean. Finally, although Ωα proves to be very intricate and unmanageable for α∈[g2,(10-2)/3), the α-Legendre constant L(α) on this interval is explicitly given.