Natural extensions for Nakada's α-expansions: Descending from 1 to g&lt;sup&gt;2&lt;/sup&gt;

de Jonge, C.J.; Kraaikamp, C.

doi:10.1016/j.jnt.2017.07.012

Natural extensions for Nakada's α-expansions

Title

Natural extensions for Nakada's α-expansions: Descending from 1 to g²

Author

de Jonge, C.J. (TU Delft Applied Probability; Universiteit van Amsterdam)
Kraaikamp, C. (TU Delft Applied Probability)

Date

2018-02

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.

Subject

Continued fractions
Metric theory

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http://resolver.tudelft.nl/uuid:c61d1b76-deac-4589-a6aa-a3f6e11ec256

DOI

https://doi.org/10.1016/j.jnt.2017.07.012

Embargo date

2019-10-23

ISSN

0022-314X

Source

Journal of Number Theory, 183, 172-212

Bibliographical note

Accepted Author Manuscript

Part of collection

Institutional Repository

Document type

journal article

Rights

Files

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