Natural Extensions for Nakada's alpha-expansions: descending from 1 to g^2

TL;DR

This paper constructs the natural extension of Nakada's alpha-expansions for a specific alpha range, revealing isomorphisms with golden mean related domains and explicitly determining the alpha-Legendre constant.

Contribution

It introduces a method using singularisations and insertions to extend Nakada's alpha-expansions and characterizes the natural extension domains for a range of alpha values.

Findings

The natural extension domain mma_lpha is isomorphic to mma_g for certain alpha.

Explicit formula for the lpha-Legendre constant on the interval.

Construction of the natural extension for lpha in a new parameter range.

Abstract

By means of singularisations and insertions in Nakada's alpha-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 T_alpha is given for (\sqrt{10}-2)/3\leq\alpha<1. From our construction it follows that \Omega_\alpha, the domain of the natural extension of T_\alpha, is metrically isomorphic to \Omega_g for \alpha \in [g^2,g), where g is the small golden mean. Finally, although \Omega_\alpha proves to be very intricate and unmanageable for \alpha \in [g^2, (\sqrt{10}-2)/3), the \alpha-Legendre constant L(\alpha) on this interval is explicitly given.