Orbits of N-expansions with a finite set of digits

Abstract

AbstractFor$$N \in {\mathbb {N}}_{\ge 2}$$N∈N≥2and$$\alpha \in {\mathbb {R}}$$α∈Rsuch that$$0 < \alpha \le \sqrt{N}-1$$0<α≤N-1, we define$$I_\alpha :=[\alpha ,\alpha +1]$$Iα:=[α,α+1]and$$I_\alpha ^-:=[\alpha ,\alpha +1)$$Iα-:=[α,α+1)and investigate the continued fraction map$$T_{\alpha }:I_{\alpha }\rightarrow I_{\alpha }^-$$Tα:Iα→Iα-, which is defined as$$T_{\alpha }(x):= \frac{N}{x}-d(x),$$Tα(x):=Nx-d(x),where$$d: I_{\alpha }\rightarrow {\mathbb {N}}$$d:Iα→Nis defined by$$d(x):=\left\lfloor \frac{N}{x} -\alpha \right\rfloor $$d(x):=Nx-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)⊂Iαexist such that for almost every$$x \in I_{\alpha }$$x∈Iαthere is an$$n_0 \in {\mathbb {N}}$$n0∈Nfor which$$T_{\alpha }^n(x)\notin (a,b)$$Tαn(x)∉(a,b)for all$$n\ge n_0$$n≥n0. Thesegaps(a, b) are investigated using the square$$\varUpsilon _\alpha :=I_{\alpha }\times I_{\alpha }^-$$Υα:=Iα×Iα-, where theorbits$$T_{\alpha }^k(x), k=0,1,2,\ldots $$Tαk(x),k=0,1,2,…of numbers$$x \in I_{\alpha }$$x∈Iαare represented as cobwebs. The squares$$\varUpsilon _\alpha $$Υαare the union offundamental regions, which are related to the cylinder sets of the map$$T_{\alpha }$$Tα, according to the finitely many values ofdin$$T_{\alpha }$$Tα. In this paper some clear conditions are found under which$$I_{\alpha }$$Iαis gapless. If$$I_{\alpha }$$Iαconsists 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$$I_{\alpha }$$Iαwith regard to the fixed points of$$I_{\alpha }$$Iαunder$$T_{\alpha }$$Tα.