The $\beta$-transformation with a hole

This paper extends those of Glendinning and Sidorov [3] and of Hare and Sidorov [6] from the case of the doubling map to the more general $\beta$-transformation. Let $\beta \in (1,2)$ and consider the $\beta$-transformation $T_{\beta}(x)=\beta x \pmod 1$. Let $\mathcal{J}_{\beta} (a,b) := \{ x \in (0,1) : T_{\beta}^n(x) \notin (a,b) \text{ for all } n \geq 0 \}$. An integer $n$ is bad for $(a,b)$ if every $n$-cycle for $T_{\beta}$ intersects $(a,b)$. Denote the set of all bad $n$ for $(a,b)$ by $B_\beta(a,b)$. In this paper we completely describe the following sets: $$D_0(\beta) = \{ (a,b) \in [0,1)^2 : \mathcal{J}_{\beta}(a,b) \neq \emptyset \},$$ $$D_1(\beta) = \{ (a,b) \in [0,1)^2 : \mathcal{J}_{\beta}(a,b) \text{ is uncountable} \},$$ $$D_2(\beta) = \{ (a,b) \in [0,1)^2 : B_\beta(a,b) \text{ is finite} \}.$$