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

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\leqα<1. From our construction it follows that Ω_α, the domain of the natural extension of T_α, is metrically isomorphic to Ω_g for α\in [g^2,g), where g is the small golden mean. Finally, although Ω_αproves to be very intricate and unmanageable for α\in [g^2, (\sqrt{10}-2)/3), the α-Legendre constant L(α) on this interval is explicitly given.