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

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.