Natural Extensions for Nakada's alpha-expansions: descending from 1 to g2
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 Talpha is given for (10-2)/3≤α<1. From our construction it follows that α, the domain of the natural extension of Tα, is metrically isomorphic to g for α ∈ [g2,g), where g is the small golden mean. Finally, although α 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.
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