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

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 ∈ [g 2 , g), where g is the small golden mean. Finally, although Ωα proves to be very intricate and unmanageable for α ∈ [g 2 , (√ 10 − 2)/3), the α-Legendre constant L(α) on this interval is explicitly given.

α-Expansions with odd partial quotients

“…This interpolates between the regular continued fractions (RCF) when α = 1 and the nearest integer continued fractions (NICF) when α = 1/2. Kraaikamp [7] investigated additional properties of this α-expansion in the range α ∈ [1/2, 1], and several other works analyzed the more challenging situation where α < 1/2 (see, e.g., [1,3,6,8,9,10,12,13,19]). Nakada's α-expansions became part of a broader class of continued fractions.…”