Nakada α-展开式的自然扩张：从1降到g^2
Natural Extensions for Nakada's alpha-expansions: descending from 1 to g^2
摘要：
利用Nakada α-展开式中的奇异化和插入，即去掉部分分式1而引入带减号的部分分式，给出了（\sqrt{10} -2）/3 \leq α<1时Nakada连分数映射T α的自然扩张.从我们的构造可以得出，T_α的自然扩张的定义域Ω_α与Ω_g度量同构，其中α \in [g^2，g）是小黄金平均.最后，虽然Ω_α对于α \in [g^2，（\sqrt{10} -2）/3）证明是非常复杂和难以处理的，但在这个区间上的α-Legendre常数L（α）是明确给出的.
英文摘要：
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.
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