The $β$-transformation with a hole (1412.6384v3)

Abstract: 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} }. ]