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Cited by 29 publications
References 13 publications
A new continued fraction expansion algorithm, the so-called a/b-expansion, is introduced and some of its basic properties, such as convergence of the algorithm and ergodicity of the underlying dynamical system, have been obtained. Although seemingly a minor variation of the regular continued fraction (RCF) expansion and its many variants (such as Nakada's α-expansions, Schweiger's odd-and even-continued fraction expansions, and the Rosen fractions), these a/b-expansions behave very differently from the RCF and many important question remains open, such as the exact form of the invariant measure, and the "shape" of the natural extension.
A new continued fraction expansion algorithm, the so-called a/b-expansion, is introduced and some of its basic properties, such as convergence of the algorithm and ergodicity of the underlying dynamical system, have been obtained. Although seemingly a minor variation of the regular continued fraction (RCF) expansion and its many variants (such as Nakada's α-expansions, Schweiger's odd-and even-continued fraction expansions, and the Rosen fractions), these a/b-expansions behave very differently from the RCF and many important question remains open, such as the exact form of the invariant measure, and the "shape" of the natural extension.
We generalize the result of Wirsing on Gauss transformation to the generalized tranformation Tp(x) = { p x } for any positive integer p. We give an estimate for the generalized Gauss-Kuzmin-Wirsing constant.
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 .
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