Orbits of N-expansions with a finite set of digits

Abstract For $$N \in {\mathbb {N}}_{\ge 2}$$ N∈N≥2 and $$\alpha {R}}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">α∈R such that $$0 < \alpha \le \sqrt{N}-1$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">0<α≤N-1 , we define $$I_\alpha :=[\alpha ,\alpha +1]$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Iα:=[α,α+1] ^-:=[\alpha +1)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Iα-:=[α,α+1) investigate the continued fraction map $$T_{\alpha }:I_{\alpha }\rightarrow I_{\alpha }^-$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Tα:Iα→Iα- which is defined as }(x):= \frac{N}{x}-d(x),$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Tα(x):=Nx-d(x), where $$d: {N}}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">d:Iα→N by $$d(x):=\left\lfloor \frac{N}{x} -\alpha \right\rfloor $$ xmlns:mml="http://www.w3.org/1998/Math/MathML">d(x):=Nx-α . $$N\in 7}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">N∈N≥7 for certain values of xmlns:mml="http://www.w3.org/1998/Math/MathML">α open intervals $$(a,b) \subset }$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">(a,b)⊂Iα exist almost every $$x xmlns:mml="http://www.w3.org/1998/Math/MathML">x∈Iα there an $$n_0 xmlns:mml="http://www.w3.org/1998/Math/MathML">n0∈N }^n(x)\notin (a,b)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Tαn(x)∉(a,b) all $$n\ge n_0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">n≥n0 These gaps ( a b ) are investigated using square $$\varUpsilon _\alpha :=I_{\alpha }\times xmlns:mml="http://www.w3.org/1998/Math/MathML">Υα:=Iα×Iα- orbits }^k(x), k=0,1,2,\ldots xmlns:mml="http://www.w3.org/1998/Math/MathML">Tαk(x),k=0,1,2,… numbers represented cobwebs. The squares xmlns:mml="http://www.w3.org/1998/Math/MathML">Υα union fundamental regions related to cylinder sets xmlns:mml="http://www.w3.org/1998/Math/MathML">Tα according finitely many d in In this paper some clear conditions found under $$I_{\alpha xmlns:mml="http://www.w3.org/1998/Math/MathML">Iα gapless. If consists at least five sets, it always case four usually no gaps, except rare cases one, very wide gap. Gaplessness two or three depends on position endpoints with regard fixed points