For $$N \in {\mathbb {N}}_{\ge 2}$$ N ∈ N ≥ 2 and $$\alpha \in {\mathbb {R}}$$ α ∈ R such that $$0 < \alpha \le \sqrt{N}-1$$ 0 < α ≤ N - 1 , we define $$I_\alpha :=[\alpha ,\alpha +1]$$ I α : = [ α , α + 1 ] and $$I_\alpha ^-:=[\alpha ,\alpha +1)$$ I α - : = [ α , α + 1 ) and investigate the continued fraction map $$T_{\alpha }:I_{\alpha }\rightarrow I_{\alpha }^-$$ T α : I α → I α - , which is defined as $$T_{\alpha }(x):= \frac{N}{x}-d(x),$$ T α ( x ) : = N x - d ( x ) , where $$d: I_{\alpha }\rightarrow {\mathbb {N}}$$ d : I α → N is defined by $$d(x):=\left\lfloor \frac{N}{x} -\alpha \right\rfloor $$ d ( x ) : = N x - α . For $$N\in {\mathbb {N}}_{\ge 7}$$ N ∈ N ≥ 7 , for certain values of $$\alpha $$ α , open intervals $$(a,b) \subset I_{\alpha }$$ ( a , b ) ⊂ I α exist such that for almost every $$x \in I_{\alpha }$$ x ∈ I α there is an $$n_0 \in {\mathbb {N}}$$ n 0 ∈ N for which $$T_{\alpha }^n(x)\notin (a,b)$$ T α n ( x ) ∉ ( a , b ) for all $$n\ge n_0$$ n ≥ n 0 . These gaps (a, b) are investigated using the square $$\varUpsilon _\alpha :=I_{\alpha }\times I_{\alpha }^-$$ Υ α : = I α × I α - , where the orbits$$T_{\alpha }^k(x), k=0,1,2,\ldots $$ T α k ( x ) , k = 0 , 1 , 2 , … of numbers $$x \in I_{\alpha }$$ x ∈ I α are represented as cobwebs. The squares $$\varUpsilon _\alpha $$ Υ α are the union of fundamental regions, which are related to the cylinder sets of the map $$T_{\alpha }$$ T α , according to the finitely many values of d in $$T_{\alpha }$$ T α . In this paper some clear conditions are found under which $$I_{\alpha }$$ I α is gapless. If $$I_{\alpha }$$ I α consists of at least five cylinder sets, it is always gapless. In the case of four cylinder sets there are usually no gaps, except for the rare cases that there is one, very wide gap. Gaplessness in the case of two or three cylinder sets depends on the position of the endpoints of $$I_{\alpha }$$ I α with regard to the fixed points of $$I_{\alpha }$$ I α under $$T_{\alpha }$$ T α .

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.

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